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This article is cited in 2 scientific papers (total in 2 papers) Geometry of Diophantine exponents O. N. Germanab a Lomonosov Moscow State University b Moscow Center for Fundamental and Applied Mathematics
Russian version:
Uspekhi Matematicheskikh Nauk, 2023, Volume 78, Issue 2(470), DOI: https://doi.org/10.4213/rm10089 
Bibliographic databases:
    1. IntroductionGiven a square with side length 1, is the length of its diagonal equal to a ratio of two integers? Can one construct a circle with the same area as this square using straightedge and compass? These questions date back to the times of Ancient Greece — about two and a half thousand years ago. The Greeks answered the first question immediately, whereas the second had remained open until only little more than a century ago. The answer to the first question proved the existence of irrational numbers. However, explicit examples of such numbers, as a rule, were algebraic, that is, they were roots of polynomials with rational coefficients. And until the 19th century, it had been unclear whether there exist non-algebraic, that is, transcendental, numbers. It was Liouville who managed to answer this question1 by showing that algebraic numbers cannot be approximated by rationals ‘too well’, after which he easily constructed an example of a transcendental number. This is how the theory of Diophantine approximation was born. It was this theory that provided a proof of the transcendence of $\pi$, and therefore an answer to the second of the two questions mentioned above — that it is impossible to construct a circle with the same area as a given square using straightedge and compass. Hence came the understanding that real numbers can be ranked by their ‘irrationality measure’ — the better an irrational number can be approximated by rationals, the more irrational it is considered. The simplest quantitative characteristic of how well an irrational number $\theta$ can be approximated by rationals, is its Diophantine exponent — the supremum of real $\gamma$ such that the inequality
$$
\begin{equation*}
\biggl|\theta-\frac{p}{q}\biggr|\leqslant\frac{1}{q^{1+\gamma}}
\end{equation*}
\notag
$$
admits infinitely many solutions in integers $p$ and $q$. The problem of approximation of a real number by rationals has a very natural geometric interpretation, which helps to work with more general problems — when we need to find so called simultaneous approximations, that is, when several numbers are to be approximated by rationals with the same denominator. In this context there also appear Diophantine exponents as the simplest measure of deviation from ‘rationality’. It is worth mentioning that the problem of simultaneous approximations to powers of a single number $\theta$, that is, to the numbers $1, \theta, \theta^2, \theta^3, \dots, \theta^n$, provides one of the most important tools for studying algebraic numbers. For instance, in his proof of the transcendence of $e$, Hermite actually constructed ‘good’ simultaneous approximations to powers of $e$. For the past several years, the theory of Diophantine approximation experienced a significant progress: the number of new theorems exceeded the number of those proved before. In this survey we tried to gather as many results as possible concerning various Diophantine exponents — and there are more than a dozen types of them nowadays. We have no claim for exhaustiveness, and we provide only the proofs that are concise enough and essential for understanding the methods of working with the objects under study. Detailed proofs can be found in the original papers. The multidimensional setting will be of most interest to us, as multidimensionality delivers a variety of natural ways to define Diophantine exponents. We pay special attention to the so-called transference principle, which connects ‘dual’ problems. We note also that we almost do not mention intermediate Diophantine exponents, since a proper account of what is currently known about them would double the size of the paper. 2. Approximation of a real number by rationals2.1. The Diophantine exponent and irrationality measureIn the introduction we actually gave the following definition. Definition 1. Let $\theta$ be a real number. Its Diophantine exponent $\omega(\theta)$ is defined as the supremum of real $\gamma$ such that the inequality admits infinitely many solutions in integers $p$ and $q$. In the case of a single number it is more traditional, however, to talk about the irrationality measure of $\theta$. Definition 2. Let $\theta$ be a real number. Its irrationality measure $\mu(\theta)$ is defined as the supremum of real $\gamma$ such that the inequality admits infinitely many solutions in coprime integers $p$ and $q$. Definition 2 differs from Definition 1 only by the absence of 1 in the exponent in (2) and by the condition that $p$ and $q$ should be coprime. Thus, for irrational $\theta$ we have Whereas, if $\theta\in\mathbb{Q}$, then $\omega(\theta)=\infty$ and $\mu(\theta)=1$ (see Proposition 1 below).Most results describing the measure of deviation of a number from rationality are usually stated in terms of $\mu(\theta)$. However, the most reasonable way to define Diophantine exponents in the multidimensional setting gives exactly $\omega(\theta)$, rather than $\mu(\theta)$, in the particular one-dimensional case. Therefore, we prefer to formulate all our statements in terms of $\omega(\theta)$. 2.2. Dirichlet’s theoremAs a rule, talks on the elements of Diophantine approximation start with mentioning the corresponding Dirichlet theorem. Let us keep to this tradition. Being rather elementary, Dirichlet’s theorem on the approximation of real numbers by rationals is also a most fundamental statement contained, in this form or another, in most of the existing theorems on Diophantine approximation. Theorem 1 (Lejeune Dirichlet, 1842). Let $\theta$ and $t$ be real numbers, $t>1$. Then there are integers $p$ and $q$ such that $0<q<t$ and The original statement of the theorem, proved in Dirichlet’s paper [1] of year 1842, is actually different. It resembles much more the statement of Corollary 1 below and it was given for a linear form in arbitrarily many variables (see also Theorems 8 and 9). However, Theorem 1 immediately follows from Dirichlet’s original argument, and it so happened that this way of formulating Dirichlet’s theorem is considered to be classical. There are two best known proofs of this theorem — an arithmetic one, involving the pigeonhole principle, and a geometric one, involving Minkowski’s convex body theorem. The latter approach is discussed in § 2.4 in more detail. For irrational $\theta$ Theorem 1 provides an estimate, trivial to a certain extent, for the Diophantine exponent of an irrational number. Corollary 1. Let $\theta$ be an irrational real number. Then there are infinitely many pairs of coprime integers $p$ and $q$ such that Corollary 2. If $\theta$ is irrational, then $\omega(\theta)\geqslant1$ and $\mu(\theta)\geqslant2$. The irrationality assumption cannot be omitted, as the following result shows. Proof. Let $\theta=a/b$, where $a,b\in\mathbb{Z}$ and $b>0$.
Then for every $k\in\mathbb{N}$ and every $\gamma\in\mathbb{R}$
$$
\begin{equation*}
\biggl|\theta-\frac{ka}{kb}\biggr|=0<\frac{1}{q^\gamma}\,,
\end{equation*}
\notag
$$
from which it immediately follows that $\omega(\theta)=\infty$.
Furthermore, for every rational $p/q$ distinct from $\theta$ we have
$$
\begin{equation*}
\biggl|\theta-\frac{p}{q}\biggr|=\frac{|aq-bp|}{bq}\geqslant\frac{1/b}{q}\,,
\end{equation*}
\notag
$$
so it is clear that for every positive $\gamma$ inequality (1) admits only a finite number of solutions. Thus, $\mu(\theta)\leqslant1$. Finally, we come to $\mu(\theta)=1$ by noticing that the linear Diophantine equation with coprime $a$ and $b$ admits infinitely many solutions. $\Box$ 2.3. Diophantine and Liouville numbersIn 1844, two years after Dirichlet’s paper was published, Liouville [2] constructed the first examples of transcendental numbers. The main idea of his proof led to the following statement, which was published in [3] in 1851. Theorem 2 (Liouville, 1844–1851). Let $\theta$ be an algebraic number of degree $n$. Then there is a positive constant $c$ depending only on $\theta$ such that for every integer $p$ and every positive integer $q$ we have Theorem 2 immediately implies an estimate for the irrationality measure (as well as for the Diophantine exponent) of an algebraic number, which generalises the estimate provided by Proposition 1. Corollary 3. If $\theta$ is an algebraic number of degree $n$, then $\mu(\theta)\leqslant n$. Correspondingly, for $n\geqslant 2$ we have $\omega(\theta)\leqslant n-1$. Hence we conclude that if $\mu(\theta)=\infty$ (which, for irrational $\theta$, is equivalent to $\omega(\theta)=\infty$), then $\theta$ cannot be algebraic; therefore, it is transcendental. Definition 3. Let $\theta$ be an irrational number. If $\omega(\theta)=\infty$, then $\theta$ is called a Liouville number. If $\omega(\theta)<\infty$, then $\theta$ is called a Diophantine number. Thus, algebraic numbers are Diophantine. But what are the possible values of their Diophantine exponents? Corollaries 2 and 3 give us an analogue of Proposition 1 for quadratic irrationalities. Proposition 2. If $\theta$ is a quadratic irrationality, that is, an algebraic number of degree $2$, then $\omega(\theta)=1$ and $\mu(\theta)=2$. For algebraic numbers of higher degree Liouville’s theorem had been improved upon successively by Thue [4], Siegel [5], Dyson [6], and Gelfond [7], until Roth [8] obtained in 1955 a result for which he was awarded the Fields Medal in 1958. Theorem 3 (Roth, 1955). If $\theta$ is an irrational algebraic number, then for every positive $\varepsilon$ the inequality has at most a finite number of solutions in integers $p$ and $q$. Corollary 4. If $\theta$ is an irrational algebraic number, then $\omega(\theta)=1$ and $\mu(\theta)=2$. Thus, irrational algebraic numbers have the smallest possible value of the Diophantine exponent. Which means that algebraic numbers are badly approximable by rationals. However, the classical notion of badly approximable numbers is slightly more subtle. It assumes that the strongest possible way to improve on Corollary 1 is by decreasing the constant. Definition 4. An irrational number $\theta$ is called badly approximable (by rationals), if there is a positive constant $c$ such that for every rational $p/q$ we have It follows from Liouville’s theorem that algebraic numbers of degree $2$ are badly approximable. It is an open question whether algebraic numbers of higher degrees are badly approximable too. To date, even Lang’s conjecture (1945) is still open. It claims that for every irrational algebraic number $\theta$ there exists $\delta>1$ such that the inequality
$$
\begin{equation*}
\biggl|\theta-\frac{p}{q}\biggr|<\frac{1}{q^2(\log q)^\delta}
\end{equation*}
\notag
$$
has at most a finite number of solutions in integers $p$ and $q$. 2.4. The geometric interpretation of Dirichlet’s theoremConsider the line $\mathcal{L}(\theta)$ in $\mathbb{R}^2$ passing through the origin and the point $(1,\theta)$. The set of points $(x,y)$ satisfying the inequality forms a ‘hyperbolic cross’ $\mathcal{H}(\theta)$ with $\mathcal{L}(\theta)$ and the $y$-axis as asymptotes. As for the set determined by the system
$$
\begin{equation*}
\begin{cases} |x|\leqslant t, \\ |\theta x-y|\leqslant \dfrac{1}{t}, \end{cases}
\end{equation*}
\notag
$$
it is a parallelogram $\mathcal{P}(\theta,t)$ with sides parallel to those two lines (see Fig. 1). Clearly, Corollary 1 to Dirichlet’s theorem states that $\mathcal{H}(\theta)$ contains infinitely many points $(x,y)\in\mathbb{Z}^2$ with non-zero $x$, and Dirichlet’s theorem itself (Theorem 1) states that $\mathcal{P}(\theta,t)$ contains a non-zero point of $\mathbb{Z}^2$ for every $t>1$. Thus formulated, Dirichlet’s theorem becomes a particular case of the following classical Minkowski convex body theorem published in [9] (also see [10] and [11]). Theorem 4 (Minkowski, 1896). Let $\mathcal{M}$ be a convex centrally symmetric closed body in $\mathbb{R}^d$ centred at the origin. Suppose that the volume of $\mathcal{M}$ is at least $2^d$. Then $\mathcal{M}$ contains a non-zero point of $\mathbb{Z}^d$. Corollary 5. Let $L_1,\dots,L_d$ be a $d$-tuple of homogeneous linear forms in $\mathbb{R}^d$ with determinant $D\ne0$, and let $\delta_1,\dots,\delta_n$ be positive numbers whose product equals $|D|$. Then there is a point $\mathbf{v}\in\mathbb{Z}^d\setminus\{\mathbf{0}\}$ such that It is worth mentioning that often Corollary 5 is used instead of Minkowski’s theorem. This corollary bears the name of Minkowski’s theorem for linear forms. Clearly, Dirichlet’s theorem requires only Corollary 5. 2.5. The relationship with continued fractionsRecall the algorithm of continued fraction expansion of a real number $\theta$. Denoting by $[\,\cdot\,]$ the integer part, let us define the sequences $(\alpha_k)$, $(a_k)$, $(p_k)$, and $(q_k)$ by
$$
\begin{equation}
\begin{aligned} \, \alpha_0=\theta,\quad \alpha_{k+1}=(\alpha_k-[\alpha_k])^{-1},\quad a_k=[\alpha_k], \\ \begin{aligned} \, p_{-2} & =0, \quad p_{-1}=1, \\ q_{-2} & =1, \quad q_{-1}=0, \end{aligned}\qquad \begin{aligned} \, p_k & =a_kp_{k-1}+p_{k-2}, \\ q_k & =a_kq_{k-1}+q_{k-2}, \end{aligned} \end{aligned}\qquad k=0,1,2,\dots\,.
\end{equation}
\tag{3}
$$
These sequences are infinite, provided that $\theta$ is irrational. If $\theta$ is rational, they stop as soon as $\alpha_k$ becomes an integer. Then (see [12] and [11]), for every $k=0,1,2,\dots$ such that $\alpha_k$ is well defined, we have
$$
\begin{equation*}
\theta=[a_0;a_1,\dots,a_{k-1},\alpha_k]= a_0+\cfrac{1}{a_1+\cfrac{1}{\stackrel{\ddots}{\phantom{|}}+ \cfrac{1}{a_{k-1}+\cfrac{1}{\alpha_k}}}}\quad\text{and}\quad \frac{p_k}{q_k}=[a_0;a_1,\dots,a_k].
\end{equation*}
\notag
$$
The numbers $a_k$ are called partial quotients of $\theta$ and the $p_k/q_k$ are called convergents of $\theta$. 2.5.1. The Diophantine exponent and growth of partial quotientsThe convergents of $\theta$ satisfy (see [11]–[13]) the relation
$$
\begin{equation*}
\biggl|\theta-\frac{p}{q}\biggr|\leqslant\frac{1}{q^2}\,.
\end{equation*}
\notag
$$
On the other hand, if a rational number $p/q$ satisfies this inequality, then by Fatou’s theorem (see [13]–[15]) $p/q$ coincides either with a convergent of $\theta$, or with an intermediate fraction neighbouring a convergent. Moreover, by Legendre’s theorem (see [11]–[13]) any rational number $p/q$ satisfying the inequality
$$
\begin{equation*}
\biggl|\theta-\frac{p}{q}\biggr|\leqslant\frac{1}{2q^2}
\end{equation*}
\notag
$$
coincides with a convergent of $\theta$. Thus, the order of approximation of $\theta$ by rationals is determined by how rapidly the differences between $\theta$ and its convergents decrease. There exist classical estimates for these differences (see [11]–[13]). Proposition 3. Let $\theta=[a_0;a_1,a_2,\dots]$ be the continued fraction expansion of $\theta$, and let $p_k/q_k=[a_0;a_1,\dots,a_k]$ be its convergents. Then Corollary 6. An irrational number is badly approximable if and only if its partial quotients are bounded. Corollary 7. Given an irrational number $\theta$, with the notation of Proposition 3 we have Thus, the order of approximation of $\theta$ by rationals is determined by the growth of the partial quotients of $\theta$. 2.5.2. A geometric algorithmThe approach described in § 2.4 enables interpreting a continued fraction as a geometric object (see also [16]–[20]). Let us define sequences of real numbers $(\beta_k)$ and $(b_k)$ and a sequence of lattice points $(\mathbf{v}_k)$. Set
$$
\begin{equation*}
\mathbf v_{-2}=(1,0)\quad\text{and}\quad \mathbf v_{-1}=(0,1).
\end{equation*}
\notag
$$
Given $\mathbf{v}_{k-2}$ and $\mathbf{v}_{k-1}$, define $\beta_k$, $b_k$, and $\mathbf{v}_k$ by
$$
\begin{equation*}
\mathbf v_{k-2}+\beta_k\mathbf v_{k-1}\in\mathcal{L}(\theta),\qquad b_k=[\beta_k],\quad\text{and}\quad \mathbf v_k=\mathbf v_{k-2}+b_k\mathbf v_{k-1}.
\end{equation*}
\notag
$$
In other words, $\mathbf{v}_k$ is the last point of $\mathbb{Z}^2$ before crossing the line $\mathcal{L}(\theta)$ on the way from $\mathbf{v}_{k-2}$ in the direction determined by2 $\mathbf{v}_{k-1}$ (see Fig. 2). It is impossible to construct the point $\mathbf{v}_k$ if and only if $\mathbf{v}_{k-1}$ is on $\mathcal{L}(\theta)$ — in this case the three sequences stop. If there are no non-zero points of $\mathbb{Z}^2$ on $\mathcal{L}(\theta)$, then the sequences are infinite. Proof. Statements (a) and (b) follow from the fact that both determinant and the relation $\mathbf{v}_k=\mathbf{v}_{k-2}+b_k\mathbf{v}_{k-1}$ are linear.
To prove (c) let us express $\beta_{k+1}$ in terms of $\beta_k$. The vectors $\mathbf{v}_{k-2}+\beta_k\mathbf{v}_{k-1}$ and $\mathbf{v}_{k-1}+\beta_{k+1}\mathbf{v}_k$ are collinear, so that from (a) and (b) we obtain
$$
\begin{equation*}
0=\det(\mathbf v_{k-2}+\beta_k\mathbf v_{k-1},\mathbf v_{k-1}+ \beta_{k+1}\mathbf v_k)=(-1)^k+(-1)^{k-1}\beta_k\beta_{k+1}+ (-1)^kb_k\beta_{k+1}.
\end{equation*}
\notag
$$
Thus, $1-\beta_{k+1}(\beta_k-b_k)=0$, that is, which coincides with the corresponding formula for $\alpha_k$ and $\alpha_{k+1}$ and gives us the induction step. Finally, we note that
$$
\begin{equation*}
\beta_0=\theta=\alpha_0,\quad \mathbf v_{-2}=(1,0)=(q_{-2},p_{-2}),\quad\text{and}\quad \mathbf v_{-1}=(0,1)=(q_{-1},p_{-1}),
\end{equation*}
\notag
$$
and make use of (3). $\Box$ Statements (a) and (b) of Theorem 5 provide a geometric interpretation of the well-known relations and Statement (a) means that for each $k$ the vectors $\mathbf{v}_{k-1}$, $\mathbf{v}_k$ form a basis of $\mathbb{Z}^2$, oriented appropriately. Statement (b) means that the integer length of the segment with endpoints $\mathbf{v}_{k-2}$ and $\mathbf{v}_k$ equals $a_k$.Definition 5. The integer length of a line segment with endpoints in $\mathbb{Z}^2$ is defined as the number of minimal segments (with respect to inclusion) with endpoints in $\mathbb{Z}^2$ which are contained in it. It easily follows from the rule for constructing $\mathbf{v}_k$ from the two preceding points that for every $k\geqslant0$ the points $\mathbf{v}_{k-1}$ and $\mathbf{v}_k$ are separated by $\mathcal{L}(\theta)$. Furthermore, the points with even non-negative indices lie below $\mathcal{L}(\theta)$, and the ones with odd indices lie above $\mathcal{L}(\theta)$. This corresponds to the fact that all convergents with even indices are smaller than $\theta$ and the ones with odd indices are greater than $\theta$. 2.5.3. Geometric proofsMany statements concerning continued fractions can be proved geometrically. For example, let us present an argument that proves inequalities (b) in Proposition 3. We rewrite them as
$$
\begin{equation}
\frac{1}{a_{k+1}+2}<q_k|\theta q_k-p_k|\leqslant\frac{1}{a_{k+1}}
\end{equation}
\tag{5}
$$
and notice that $q_k|\theta q_k-p_k|$ is just the area of the parallelogram $\mathcal{Q}_1$ with vertices $\mathbf{a}$, $\mathbf{v}_k$, $\mathbf{b}$, and $\mathbf{0}$ (see Fig. 3). Consider the parallelogram $\mathcal{Q}_2$ with vertices $\mathbf{0}$, $\mathbf{c}$, $\mathbf{d}$, and $\mathbf{e}$ (see the same Fig. 3). Proof. Since $\mathbf{v}_k$ and $\mathbf{v}_{k-1}$ form a basis of $\mathbb{Z}^2$, the width $H$ of the strip in Fig. 3 equals $|\mathbf{v}_k|^{-1}$. Denote by $h$ the distance from $\mathbf{b}$ to the line spanned by $\mathbf{v}_k$. Then it follows from the similarity of the corresponding triangles that so that In view of Lemma 1 inequalities (5) can be rewritten as that is, as
$$
\begin{equation*}
|\det(\mathbf v_{k-1},\mathbf v_{k+1})|\leqslant |\det(\mathbf e,\mathbf c)|<|\det(\mathbf v_{k-1},\mathbf v_{k+1})|+2.
\end{equation*}
\notag
$$
These inequalities immediately follow from the fact that the segment $[\mathbf{e},\mathbf{c}]$ contains the segment $[\mathbf{v}_{k-1},\mathbf{v}_{k+1}]$ and is contained in $[\mathbf{v}_{k-1}-\mathbf{v}_k,\mathbf{v}_{k+1}+\mathbf{v}_k]$. This is how statement (b) of Proposition 3 is proved geometrically. Statement (a) can be proved similarly. 2.5.4. Klein polygonsThe following theorem establishes a relationshp between continued fractions and a construction which goes back to Klein [21] and bears the name of Klein polygons. Denote by $\mathcal{O}_+$ the closure of the positive quadrant, that is, the set of points in $\mathbb{R}^2$ with non-negative coordinates. Theorem 6. Let $\theta$ be positive. Then the points $\mathbf{v}_k$ with even indices are the vertices of the convex hull of the non-zero integer points lying in $\mathcal{O}_+$ not above $\mathcal{L}(\theta)$. Similarly, the points $\mathbf{v}_k$ with odd indices are the vertices of the convex hull of the non-zero integer points lying in $\mathcal{O}_+$ not below $\mathcal{L}(\theta)$. Proof. Since $\theta>0$, the coordinates of every $\mathbf{v}_k$ are non-negative. In addition, we know that if $k$ is even, then $\mathbf{v}_k$ lies below $\mathcal{L}(\theta)$ and if $k$ is odd, then $\mathbf{v}_k$ lies above $\mathcal{L}(\theta)$. Therefore, it suffices to show that there are no non-zero points of $\mathbb{Z}^2$ in the intersection of the strip shown in Fig. 3 with the angle between $\mathcal{L}(\theta)$ and the corresponding coordinate axis, apart from those on the segment $[\mathbf{v}_{k-1},\mathbf{v}_{k+1}]$.
For $k\geqslant0$ the representation
$$
\begin{equation*}
\mathbf v_{k-1}-\mathbf v_k=\mathbf v_{k-1}-(a_k\mathbf v_{k-1}+ \mathbf v_{k-2})=(1-a_k)\mathbf v_{k-1}-\mathbf v_{k-2}
\end{equation*}
\notag
$$
implies that at least one of the coordinates of the point $\mathbf{v}_{k-1}-\mathbf{v}_k$ is negative. For $k=-1$ this is even more obvious.
Thus, for every $k$ the point $\mathbf{v}_{k-1}-\mathbf{v}_k$ does not belong to $\mathcal{O}_+$. As for the point $\mathbf{v}_{k+1}+\mathbf{v}_k$, by construction it does not belong to the closure of the angle containing the segment $[\mathbf{v}_{k-1},\mathbf{v}_{k+1}]$. Taking into account that there are no points of $\mathbb{Z}^2$ in the interior of the strip shown in Fig. 3, we complete the proof. $\Box$ Thus, the continued fraction of $\theta$ is ‘written’ on the boundaries of Klein polygons: vertices have coordinates equal to the denominators and numerators of convergents and the integer lengths of edges are equal to partial quotients. We note that we may confine ourselves to one of the two polygonal lines. The reason is that each angle formed by the segments $[\mathbf{v}_{k-2},\mathbf{v}_k]$ and $[\mathbf{v}_k,\mathbf{v}_{k+2}]$ can be equipped (see [20], [22], and [23]) with its ‘integer value’ by defining the latter as the absolute value of the determinant of the shortest integer vectors parallel to these segments. In our case these are the vectors $\mathbf{v}_{k-1}$ and $\mathbf{v}_{k+1}$. As we know (see statement (b) of Theorem 5), the absolute value of their determinant is equal to the partial quotient $a_{k+1}$. Thus, each of the two polygonal lines contains all information concerning the continued fraction of $\theta$. 2.5.5. Best approximationsThe convergents are best approximations of $\theta$ (see [24]) in the following sense. Definition 7. A rational number $p/q$ is called a best approximation of $\theta$ if $p$ is the closest integer to $q\theta$ and for all rational numbers $p'/q'$ such that $q'<q$ we have Geometrically, this means that there are no non-zero points of $\mathbb{Z}^2$ in the parallelogram centred at the origin with a vertex at the point $(q,p)$ and sides parallel to $\mathcal{L}(\theta)$ and the $y$-axis, apart from its vertices (the remaining two vertices can also belong to $\mathbb{Z}^2$: this occurs if and only if $\theta\in\frac{1}{2}\mathbb{Z}\setminus\mathbb{Z}$). Klein polygons provide a rather simple proof of the fact that if $\theta$ is not a half-integer, then the set of its best approximations coincides exactly with the set of its convergents. Theorem 7. Let $\theta\in\mathbb{R}\setminus \bigl(\frac{1}{2}\mathbb{Z}\bigr)$. Then the vertices $\mathbf{v}_k$, $k\geqslant0$, of the Klein polygons correspond to the best approximations of $\theta$. Proof. Let $\mathcal{Q}_{\mathbf{v}}$ be the parallelogram centred at the origin with a vertex at the prescribed point $\mathbf{v}$ and sides parallel to $\mathcal{L}(\theta)$ and to the $y$-axis. Let $\Delta_{\mathbf{v}}$ be the half of $\mathcal{Q}_{\mathbf{v}}$ shifted by the vector $\mathbf{v}$ (as shown in Fig. 5). Then, since $\mathbb{Z}^2$ is closed under addition, the following equivalencies hold:
$$
\begin{equation*}
\begin{aligned} \, \mathcal{Q}_{\mathbf v}\cap\mathbb{Z}^2=\{\mathbf 0,\pm\mathbf v\} & \ \ \Longleftrightarrow\ \ (\mathcal{Q}_{\mathbf v}+\mathbf v)\cap \mathbb{Z}^2=\{\mathbf 0,\mathbf v,2\mathbf v\} \\ & \ \ \Longleftrightarrow\ \ \Delta_{\mathbf v}\cap \mathbb{Z}^2=\{\mathbf 0,\mathbf v\}. \end{aligned}
\end{equation*}
\notag
$$
For $\theta$ that is not a half-integer, the first of these three statements is equivalent to the fact that $\mathbf{v}$ corresponds to a best approximation of $\theta$. The last one corresponds to the fact that $\mathbf{v}$ is a vertex of the corresponding Klein polygon. $\Box$ 2.6. The triviality of the uniform analogue of the Diophantine exponentInequality (1) admits infinitely many solutions if and only if there are arbitrarily large values of $t$ such that the system
$$
\begin{equation}
\begin{cases} 0<q\leqslant t, \\ |\theta q-p|\leqslant t^{-\gamma} \end{cases}
\end{equation}
\tag{6}
$$
has a solution in integers $p$ and $q$. This is exactly what Corollary 1 to Dirichlet’s theorem claims for $\gamma=1$ (provided that $\theta$ is irrational). Which gives the corresponding bound for $\omega(\theta)$. But Dirichlet’s theorem itself (Theorem 1) claims for $\gamma=1$ that (6) has a solution for every $t$ large enough. Therefore, it is natural to consider the following uniform analogue of $\omega(\theta)$:
$$
\begin{equation*}
\begin{aligned} \, \widehat\omega(\theta)=\sup\{\gamma\in\mathbb{R}\mid &\text{ there exists } T\in\mathbb{R}\text{ such that for all } t\geqslant T \\ &\text{ the system (6) has a solution in }(p,q)\in\mathbb{Z}^2\}. \end{aligned}
\end{equation*}
\notag
$$
Dirichlet’s theorem implies the inequality It appears that in the case of a single number the uniform exponent is almost trivial: for rational $\theta$ we have (see the proof of Proposition 1), and for irrational $\theta$, in view of Proposition 4, which we are about to prove, the inequality sign in (7) can be replaced by equality sign.Proposition 4. If $\theta$ is irrational, then there are arbitrarily large values of $t$ such that the system has no solutions in integers $p$ and $q$. Proof. Consider an arbitrary vertex $\mathbf{v}$ of one of the Klein polygons. By Theorem 7 it corresponds to some best approximation of $\theta$, that is, the parallelogram $\mathcal{Q}_{\mathbf{v}}$ (see Figs. 5 and 6) contains no integer points distinct from $\mathbf{0}$ and $\pm\mathbf{v}$.
Let us extend $\mathcal{Q}_{\mathbf{v}}$ along $\mathcal{L}(\theta)$ until we meet an integer point $\mathbf{w}$ distinct from $\mathbf{0}$, $\pm\mathbf{v}$. Such a point does exist by Minkowski’s convex body theorem (Theorem 4). We obtain a parallelogram $\mathcal{Q}'_{\mathbf{v}}$ with vertices at points $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$, and $\mathbf{d}$ (see Fig. 6). Actually, $\mathbf{v}$ and $\mathbf{w}$ correspond to two consecutive convergents of $\theta$, but our argument does not require this fact. It suffices to observe that $\mathbf{v}$ and $\mathbf{w}$ are not collinear. Their non-collinearity implies that the area of the triangle with vertices $\mathbf{0}$, $\mathbf{v}$, and $\mathbf{w}$ is not less than $1/2$. Hence the area of $\mathcal{Q}'_{\mathbf{v}}$ is greater than $2$. Thus, for $t$ equal to the abscissa of $\mathbf{w}$ system (8) has no solution in integers $p$ and $q$. We can apply the extension procedure to $\mathcal{Q}_{\mathbf{w}}$ and obtain the next ‘empty’ parallelogram. This procedure can be repeated infinitely many times, since the line $\mathcal{L}(\theta)$ contains no non-zero integer points by the irrationality of $\theta$. Hence the abscissa of $\mathbf{w}$ can attain arbitrarily large values. $\Box$ Thus, considering $\widehat\omega(\theta)$ for real $\theta$ is quite meaningless because this quantity is degenerate. This is, however, not the case in the multidimensional setting, to which we now proceed. 3. Simultaneous approximation and approximation of zero by values of a linear form3.1. Two more theorems of DirichletIn the previous section our main question was how small the quantity can be for integers $x$ and $y$. The multidimensional case, that is, the case of $n$ numbers $\theta_1,\dots,\theta_n$ instead of one number $\theta$, admits two ways to generalise this problem. We can study how small the quantities can be, or we can study how small the values of the linear form can be.The first multidimensional result in Diophantine approximation belongs to Dirichlet and refers to both of these questions. In the paper [1] of year 1842 mentioned above he proved the following statement. Theorem 8 (Lejeune Dirichlet, 1842). Let $\Theta=(\theta_1,\dots,\theta_n)\in\mathbb{R}^n$. Then for each $t\geqslant1$, there are integers $x_1,\dots,x_n,y$ satisfying the inequalities It is worth making a remark similar to the one we made right after Theorem 1: Dirichlet’s original statement is weaker than the statement of Theorem 8, but his argument proves Theorem 8 as well. In the same paper Dirichlet generalised his result to the case of several linear forms. For the forms $\theta_1x-y_1,\dots,\theta_nx-y_n$, this generalisation reads as follows. Theorem 9 (Lejeune Dirichlet, 1842). Let $\Theta=(\theta_1,\dots,\theta_n)\in\mathbb{R}^n$. Then for each $t\geqslant1$ there are integers $x,y_1,\dots,y_n$ satisfying the inequalities Thus, Theorem 8 provides an estimate for the order of approximation of zero by the values of the linear form
$$
\begin{equation*}
L_\Theta(x_1,\dots,x_n,y)=\theta_1x_1+\cdots+\theta_nx_n-y
\end{equation*}
\notag
$$
at integers $x_1,\dots,x_n,y$, and Theorem 9 provides an estimate for the order of simultaneous approximation of the numbers $\theta_1,\dots,\theta_n$ by rationals with the same denominator. This is how we obtain the first estimates for the Diophantine exponents of the $n$-tuple $\Theta$ and of the linear form $L_\Theta$, whose definitions are given in § 3.2. It is worth noting that, just as Theorem 1, Theorems 8 and 9 follow immediately from Minkowski’s convex body theorem in the form of Corollary 5 as applied to systems (9) and (10). 3.2. Regular and uniform exponentsDefinition 8. The supremum of real $\gamma$ satisfying the condition that there exist arbitrarily large $t$ such that the system admits a solution in integers $x,y_1,\dots,y_n$ is called the regular Diophantine exponent of $\Theta$ and is denoted by $\omega(\Theta)$. Definition 9. The supremum of real $\gamma$ satisfying the condition that there exist arbitrarily large $t$ such that the system admits a solution in integers $x_1,\dots,x_n,y$ is called the regular Diophantine exponent of $L_\Theta$ and is denoted by $\omega(L_\Theta)$. Replacing the words ‘there exist arbitrarily large $t$ such that’ by ‘for every $t$ large enough’, we get uniform analogues of regular exponents. Definition 10. The supremum of real $\gamma$ satisfying the condition that for every $t$ large enough system (11) admits a solution in integers $x,y_1,\dots,y_n$ is called the uniform Diophantine exponent of $\Theta$ and is denoted by $\widehat\omega(\Theta)$. Definition 11. The supremum of real $\gamma$ satisfying the condition that for every $t$ large enough system (12) admits a solution in integers $x_1,\dots,x_n,y$ is called the uniform Diophantine exponent of $L_\Theta$ and is denoted by $\widehat\omega(L_\Theta)$. As we said before, Theorems 8 and 9 provide ‘trivial’ bounds for these four exponents:
$$
\begin{equation}
\omega(\Theta)\geqslant\widehat\omega(\Theta)\geqslant\frac{1}{n}\quad\text{and}\quad \omega(L_\Theta)\geqslant\widehat\omega(L_\Theta)\geqslant n.
\end{equation}
\tag{13}
$$
These inequalities are sharp in the sense that there exist $n$-tuples $\Theta$ for which each of them turns to equality. For instance, Perron [25] proved the following statement. Theorem 10 (Perron, 1921). Let $\theta_1,\dots,\theta_n$ be elements of a real algebraic number field of degree ${n+1}$, linearly independent with the unity over $\mathbb{Q}$. Then there is a positive constant $c$ depending only on $\theta_1,\dots,\theta_n$ such that the inequality admits only finitely many solutions in integers $x,y_1,\dots,y_n$. Corollary 9. If $\Theta$ is as in Theorem 10, then $\omega(\Theta)=\widehat\omega(\Theta)=1/n$. Particularly, for $\Theta=(\theta,\theta^2,\dots,\theta^n)$, where $\theta$ is a real algebraic number of degree ${n+1}$, we have $\omega(\Theta)=1/n$. In his proof of Theorem 10 Perron used an argument which also proves the implication
$$
\begin{equation}
\omega(L_\Theta)=n\ \ \Longrightarrow\ \ \omega(\Theta)=\frac{1}{n}\,.
\end{equation}
\tag{14}
$$
The reverse implication
$$
\begin{equation}
\omega(\Theta)=\frac{1}{n}\ \ \Longrightarrow\ \ \omega(L_\Theta)=n
\end{equation}
\tag{15}
$$
follows from the proof of the main result in Khintchine’s paper [26] (see also the collection of selected papers by Khintchine [27]). This gives us an analogue of Corollary 9. Corollary 10. If $\Theta$ ia as in Theorem 10, then $\omega(L_\Theta)=\widehat\omega(L_\Theta)=n$. In contrast to the case $n=1$, the uniform exponents $\widehat\omega(\Theta)$ and $\widehat\omega(L_\Theta)$ are no longer trivial for $n\geqslant2$. They can attain finite values strictly greater than $1/n$ and $n$, respectively. The existence of linear forms $L_\Theta$ with $\widehat\omega(L_\Theta)=+\infty$ that do not vanish at non-zero integer points, as well as the existence of $\Theta$ with $\widehat\omega(\Theta)=1$, was proved by Khintchine in 1926 in one of his most famous papers [28]. Such $n$-tuples and linear forms belong to a slightly wider class of Khintchine’s singular systems (see Khintchine’s papers [29] and [30], and also the remarkable survey [31] by Moshchevitin). Finite values strictly greater than $1$ cannot be attained by $\widehat\omega(\Theta)$. The reason is the same effect that implies the triviality of the uniform exponent in the case $n=1$. Indeed, let $\Theta=(\theta_1,\dots,\theta_n)\in\mathbb{R}^n\setminus\mathbb{Q}^n$, and let $\varepsilon>0$ be fixed. Then for irrational $\theta_i$ Proposition 4 guarantees the existence of arbitrarily large $t$ such that the system
$$
\begin{equation*}
\begin{cases} 0<x<t, \\ |\theta_ix-y_i|<\dfrac{1}{2t} \end{cases}
\end{equation*}
\notag
$$
has no solution in integers $x$ and $y_i$. If $t$ is large enough, then the system
$$
\begin{equation*}
\begin{cases} 0<x\leqslant t, \\ \displaystyle\max_{1\leqslant i\leqslant n}|\theta_ix-y_i|\leqslant t^{-1-\varepsilon} \end{cases}
\end{equation*}
\notag
$$
has no solution in integers $x$ and $y_1,\dots,y_n$ either. Thus, for every $\Theta=(\theta_1,\dots,\theta_n) \in\mathbb{R}^n\setminus\mathbb{Q}^n$ we have 3.3. Transference principlePerron and Khintchine proved actually something stronger than (14) and (15). Their constructions have a ‘local’ nature, and therefore they proved both the equivalence
$$
\begin{equation}
\omega(\Theta)=\frac{1}{n}\ \ \Longleftrightarrow\ \ \omega(L_\Theta)=n
\end{equation}
\tag{17}
$$
and the fact that $\Theta$ and $L_\Theta$ are simultaneously badly approximable. Definition 12. An $n$-tuple $\Theta=(\theta_1,\dots,\theta_n)$ is called badly approximable if there is a positive constant $c$ depending only on $\Theta$ such that for every tuple of integers $x$, $y_1,\dots,y_n$ such that $x\ne0$ we have Definition 13. A linear form $L_\Theta(x_1,\dots,x_n,y)$ is called badly approximable if there is a positive constant $c$ depending only on $\Theta$ such that for every tuple of integers $x_1,\dots,x_n$, $y$ such that $x_1,\dots,x_n$ are not all equal to zero we have Apparently, Perron’s and Khintchine’s constructions were the first examples of exploiting the ‘duality’ of the problem of simultaneous approximation and the problem of approximating zero by values of a linear form. We discuss the essence of this phenomenon in detail in § 3.4. 3.3.1. Inequalities for regular exponentsThe first truly outstanding result relating the problem of simultaneous approximation to the problem of approximating zero by values of a linear form is Khintchine’s theorem, which was also proved in the paper [28] mentioned above. In the same paper he named the phenomenon he had discovered the transference principle. As Jarník [32], [33] showed, inequalities (18) are sharp in the following sense: for every $\gamma\in[n,+\infty]$ there are two $n$-tuples $\Theta'$ and $\Theta''$ such that
$$
\begin{equation*}
\omega(L_{\Theta'})=\omega(L_{\Theta''})=\gamma,\qquad \frac{1+\gamma}{1+\omega(\Theta')}=n,\quad\text{and}\quad \frac{1+\gamma^{-1}}{1+\omega(\Theta'')^{-1}}=\frac{1}{n}\,.
\end{equation*}
\notag
$$
3.3.2. Inequalities for uniform exponentsThe method used to prove inequalities (18) enables proving the same inequalities for the uniform exponents $\widehat\omega(\Theta)$ and $\widehat\omega(L_\Theta)$:
$$
\begin{equation}
\frac{1+\widehat\omega(L_\Theta)}{1+\widehat\omega(\Theta)}\geqslant n\quad\text{and}\quad \frac{1+\widehat\omega(L_\Theta)^{-1}}{1+\widehat\omega(\Theta)^{-1}} \geqslant \frac{1}{n}\,.
\end{equation}
\tag{19}
$$
However, these exponents satisfy stronger inequalities. For $n=2$ Jarník presented a surprising fact in his paper [34]: in this case the uniform exponents are connected by an equality. Theorem 12 (Jarník, 1938). If $n=2$ and the components of $\Theta$ are linearly independent with the unity over $\mathbb{Q}$, then Subsequently, in 1948 Khintchine [35] published a rather simple proof of Theorem 12. For $n\geqslant3$, in the same paper [34] Jarník proved that if $\theta_1,\dots,\theta_n$ are linearly independent with the unity over $\mathbb{Q}$, then, along with (19), the following inequalities hold:
$$
\begin{equation}
\begin{alignedat}{2} \widehat\omega(\Theta)&\geqslant\frac{1}{n-1} \biggl(1-\frac{1}{\widehat\omega(L_\Theta)-2n+4}\biggr)&&\quad\text{if}\quad \widehat\omega(L_\Theta)>n(2n-3), \\ \widehat\omega(\Theta)&\leqslant 1-\frac{1}{\widehat\omega(L_\Theta)-n+2} &&\quad \text{if}\quad \widehat\omega(\Theta)>\frac{n-1}{n}\,. \end{alignedat}
\end{equation}
\tag{21}
$$
In 2012 inequalities (19) and (21) were improved by this author in [36] and [37]. Theorem 13 (German, 2012). For every $\Theta=(\theta_1,\dots,\theta_n) \in\mathbb{R}^n\setminus\mathbb{Q}^n$ we have It was shown by Marnat [38] and (independently) Schmidt and Summerer [39] that both inequalities (22) are sharp. More specifically, they showed that for every $\gamma\in[n,+\infty]$ and every
$$
\begin{equation}
\delta\in\biggl[\frac{1-\gamma^{-1}}{n-1}\,,1-\frac{n-1}{\gamma}\biggr]
\end{equation}
\tag{23}
$$
there are continuum many tuples $\Theta$ whose components are linearly independent with the unity over $\mathbb{Q}$ and such that $\widehat\omega(L_{\Theta})=\gamma$ and $\widehat\omega(\Theta)=\delta$ (we note that, if $\gamma\geqslant n$, then the segment in (23) is well defined and non-empty). 3.3.3. ‘Mixed’ inequalitiesDespite the fact that inequalities (18) for the regular exponents are sharp, they can be improved if the uniform exponents are taken into account. For the first time this was done by Laurent and Bugeaud in [40] and [41]. They proved the following. Theorem 14 (Laurent and Bugeaud, 2009). If the components of $\Theta$ are linearly independent with the unity over $\mathbb{Q}$, then Laurent [42] proved that for $n=2$ these inequalities are sharp. However, for $n\geqslant3$ it was shown by Schleischitz [43] that inequalities (24) are no longer sharp. This is not surprising in the light of the following inequalities, obtained in 2013 by Schmidt and Summerer [44] (also see [45] and [46], where shorter proofs of their theorem are proposed). Theorem 15 (Schmidt and Summerer, 2013). If the components of $\Theta$ are linearly independent with the unity over $\mathbb{Q}$, then It is easy to see that inequalities (25) and (22) immediately imply inequalities (24). Thus, since Khintchine’s inequalities follow from those of Laurent and Bugeaud, all the currently known transference inequalities which connect $\omega(\Theta)$ and $\widehat\omega(\Theta)$ with $\omega(L_\Theta)$ and $\widehat\omega(L_\Theta)$, are implied by inequalities (22) and (25) (and, of course, by the ‘trivial’ inequalities (13) and (16)). 3.3.4. Inequalities between regular and uniform exponentsThere is another series of inequalities which, technically, cannot be classified as transference inequalities, as they connect regular and uniform exponents within the framework of one of the two problems under consideration, the problem of simultaneous approximation and the problem of approximating zero by values of a linear form. Nevertheless, it seems quite reasonable to place them alongside transference inequalities, since the transference inequalities themselves provide rather non-trivial relations of this kind (see Theorem 20 below). In 1950s Jarník discovered that if $\widehat\omega(\Theta)$ is large, then $\omega(\Theta)$ cannot be too small. In [47] and [48] he published the following estimates for ${n=2}$. Theorem 16 (Jarník, 1954). If $n=2$ and the components of $\Theta$ are linearly independent with the unity over $\mathbb{Q}$, then These inequalities are sharp. Their sharpness was proved by Laurent [42]. Jarník [48] also obtained inequalities in higher dimensions. He showed that the second inequality in (26) holds for every $n\geqslant2$ and that
$$
\begin{equation*}
\frac{\omega(L_\Theta)}{\widehat\omega(L_\Theta)}\geqslant \widehat\omega(L_\Theta)^{1/(n-1)}-3,
\end{equation*}
\notag
$$
provided that $\omega(\Theta)>(5n^2)^{n-1}$. In 2012, Moshchevitin [49] obtained an optimal result for $n=3$ in the problem of simultaneous approximation. Theorem 17 (Moshchevitin, 2012). If $n=3$ and the components of $\Theta$ are linearly independent with the unity over $\mathbb{Q}$, then where $G_{\rm sim}(\Theta)$ is the greatest zero of the polynomial A year later, Schmidt and Summerer [50] proved an analogue of Theorem 17 for the linear form problem. Theorem 18 (Schmidt and Summerer, 2013). If $n=3$ and the components of $\Theta$ are linearly independent with the unity over $\mathbb{Q}$, then where $G_{\rm lin}(\Theta)$ is the greatest root of the polynomial In 2018 Marnat and Moshchevitin generalised (26), (27), and (29) to the case of an arbitrary $n\geqslant2$. Their result was published in [51] in 2020. Theorem 19 (Marnat and Moshchevitin, 2020). Let $n\geqslant2$, and let the components of $\Theta$ be linearly independent with the unity over $\mathbb{Q}$. Then where $G_{\rm sim}(\Theta)$ and $G_{\rm lin}(\Theta)$ are the greatest roots of the polynomials respectively. In the same paper [51], Marnat and Moshchevitin showed that their inequalities (31) are sharp. A year later, a slightly different proof of Theorem 19 was proposed by Rivard-Cooke in his PhD thesis [52] (also see papers [53] and [54]). Note that using the notation $G_{\rm sim}$ and $G_{\rm lin}$ both in Theorems 17, 18, and in Theorem 19 is correct, since for $n=3$ the first (second) polynomial in (32) equals the first (respectively, second) polynomial in (28) multiplied by $x-1$ (respectively, by $\widehat\omega(L_\Theta)^{-1}(x-1)$). It is easy to verify with the help of (20) that for $n=2$ the right-hand sides of inequalities (26) coincide and are equal to
$$
\begin{equation*}
\frac{1-\widehat\omega(L_\Theta)^{-1}}{1-\widehat\omega(\Theta)}\,.
\end{equation*}
\notag
$$
It is interesting that Schmidt–Summerer’s inequalities (25) imply that just this expression bounds below the ratios $\omega(\Theta)/\widehat\omega(\Theta)$ and $\omega(L_\Theta)/\widehat\omega(L_\Theta)$ for every $n\geqslant2$. This can be observed from the following result obtained in [46]. Theorem 20 (German and Moshchevitin, 2022). Let $n\geqslant2$ and let the components of $\Theta$ be linearly independent with the unity over $\mathbb{Q}$. Define $G_{\rm sim}(\Theta)$ and $G_{\rm lin}(\Theta)$ in the same way as in Theorem 19. Then Another interesting corollary to Theorem 20 is the fact that Schmidt–Summerer’s inequalities (25) imply at least one of Marnat–Moshchevitin’s inequalities (31) — the one corresponding to the smallest number among $G_{\rm sim}(\Theta)$ and $G_{\rm lin}(\Theta)$. 3.4. Ideas and methods3.4.1. Mahler’s methodTen years after Khintchine published Theorem 11 describing the transference principle, Mahler [55] found a very simple proof for it, which demonstrated vividly that the problem of simultaneous approximation is ‘dual’ to the problem of approximating zero by the values of a linear form. Mahler himself called his method arithmetic (see [55]), and though he exploited Minkowski’s convex body theorem, he applied it in the form of Corollary 5, which allows one not to go deep into geometry. Let us explain his argument in [55], where he proved the right-hand inequality in (18), that is, the inequality
$$
\begin{equation}
\omega(\Theta)\geqslant\frac{\omega(L_\Theta)}{(n-1)\omega(L_\Theta)+n}\,.
\end{equation}
\tag{34}
$$
To this end let us ‘embed’ the problem of simultaneous approximation and the problem of approximating zero by values of a linear form into the same $(n+ 1)$- dimensional Euclidean space. Let $u_1,\dots,u_{n+1}$ be the Cartesian coordinates in $\mathbb{R}^{n+1}$. Let us identify the variables $x$ and $y_1,\dots,y_n$ with $u_1,\dots,u_{n+1}$, respectively, and the variables $x_1,\dots,x_n$, $y$ with $u_2,\dots,u_{n+1}$, $-u_1$, respectively. By analogy with § 2.4 we denote by $\mathcal{L}=\mathcal{L}(\Theta)$ the one-dimensional subspace with the generating vector $(1,\theta_1,\dots,\theta_n)$, and by $\mathcal{L}^\perp$ the orthogonal complement to $\mathcal{L}$. Then $\mathcal{L}$ and $\mathcal{L}^\perp$ coincide with the spaces of solutions of the equations
$$
\begin{equation*}
\max_{1\leqslant i\leqslant n}|\theta_ix-y_i|=0\quad\text{and}\quad L_\Theta(x_1,\dots,x_n,y)=0,
\end{equation*}
\notag
$$
respectively. Assuming that there exist $t$ large enough, positive $\gamma$, and a non-zero point $\mathbf{v}=(v_1,\dots,v_{n+1})\in\mathbb{Z}^{n+1}$ such that
$$
\begin{equation}
\begin{cases} \displaystyle\max_{1\leqslant i\leqslant n}|v_{i+1}|\leqslant t, \\ |v_1+\theta_1v_2+\cdots+\theta_nv_{n+1}|\leqslant t^{-\gamma}, \end{cases}
\end{equation}
\tag{35}
$$
Mahler applied Minkowski’s theorem — or, more specifically, Corollary 5 — to the parallelepiped consisting of the points $\mathbf{u}=(u_1,\dots,u_{n+1})$ satisfying the inequalities
$$
\begin{equation}
\begin{cases} |v_1u_1+\cdots+v_{n+1}u_{n+1}|<1, \\ \displaystyle\max_{1\leqslant i\leqslant n} |\theta_iu_1-u_{i+1}|\leqslant|v_1+\theta_1v_2+\cdots+\theta_nv_{n+1}|^{1/n}. \end{cases}
\end{equation}
\tag{36}
$$
The determinant of the set of linear forms involved in (36) is
$$
\begin{equation*}
\det\begin{pmatrix} v_1 & \theta_1 & \theta_2 & \dots & \theta_n \\ v_2 & -1 & 0 & \dots & 0 \\ v_3 & 0 & -1 & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ v_{n+1} & 0 & 0 & \dots & -1 \end{pmatrix}=(-1)^n(v_1+\theta_1v_2+\cdots+\theta_nv_{n+1}),
\end{equation*}
\notag
$$
which is equal in absolute value to the product of the right-hand sides of (36). Therefore, by Corollary 5 system (36) admits a non-zero integer solution $\mathbf{w}=(w_1,\dots,w_{n+1})$. Since $\mathbf{v}$ is an integer, it follows from the first inequality (36) that so that
$$
\begin{equation*}
w_1(v_1+\theta_1v_2+\cdots+\theta_nv_{n+1})=\sum_{i=1}^nv_{i+1}(\theta_iw_1-w_{i+1}).
\end{equation*}
\notag
$$
Hence by the second inequality (36)
$$
\begin{equation}
|w_1|\leqslant n\max_{1\leqslant i\leqslant n}|v_{i+1}|\, |v_1+\theta_1v_2+\cdots+\theta_nv_{n+1}|^{1/n-1}.
\end{equation}
\tag{38}
$$
Combining (35), (36), and (38) we obtain
$$
\begin{equation}
\begin{cases} |w_1|\leqslant nt^{1-\gamma(1/n-1)}= nt^{((n-1)\gamma+n)/n}=t^{((n-1)\gamma+n)/n+(\log n)/(\log t)}, \\ \displaystyle\max_{1\leqslant i\leqslant n}|\theta_iw_1-w_{i+1}| \leqslant t^{-\gamma/n}=\bigl(t^{((n-1)\gamma+n)/n+ (\log n)/(\log t)}\bigr)^{-\gamma/((n-1)\gamma+n)+o(1)}. \end{cases}
\end{equation}
\tag{39}
$$
This proves (34). The key ingredient of Mahler’s method is relation (37), which says that the required point happens to lie in the orthogonal complement to the line spanned by $\mathbf{v}$. More specifically, in the intersection of this orthogonal complement with the cylinder determined by the second inequality in (36), whose axis is the line spanned by $(1,\theta_1,\dots,\theta_n)$. Mahler generalised his method by proving in [56] his famous theorem on a bilinear form. Theorem 21 (Mahler, 1937). Assume that two sets of homogeneous linear forms are fixed in $\mathbf{u}\in\mathbb R^d$: As we will see in § 4, this theorem provides that a rather simple proof of the transference inequalities in the most general problem of homogeneous linear Diophantine approximation — when zero is to be approximated simultaneously by the values of several linear forms at integer points. We will actually reformulate it in terms of pseudocompounds (Theorem 22 below). Thus formulated, Mahler’s theorem becomes rather concise and very convenient for applications. 3.4.2. Pseudocompounds and dual latticesIn 1955, in [57] and [58] Mahler developed the theory of compound bodies (see also Gruber and Lekkerkerker’s book [59]). This theory appeared to be rather fruitful in the context of problems related to the transference principle. We reformulate Theorem 21 with the help of a construction from Schmidt’s book [11], which is a simplification of what Mahler calls the $(d-1)$st compound of a parallelepiped. Definition 14. Let $\eta_1,\dots,\eta_d$ be positive real numbers. Consider the parallelepiped Then the parallelepiped is called the $(d-1)$th pseudocompound of $\mathcal{P}$. We shall often call $\mathcal{P}^\ast$ simply the compound of $\mathcal{P}$, omitting ‘$(d-1)$th’. We also recall the definition of the dual lattice. Definition 15. Let $\Lambda$ be a full-rank lattice in $\mathbb{R}^d$. Then its dual lattice is defined by where $\langle \,\cdot\,{,}\,\cdot\,\rangle$ denotes the inner product. Note that the relation of duality is symmetric in the case of lattices, that is, Let $F$ and $G$ be the matrices from Theorem 21. Consider the lattices $F\mathbb{Z}^d$ and $G\mathbb{Z}^d$. In view of Definition 15, the fact that the coefficients of the form (40) are integers means exactly that each of these two lattices is a sublattice of the other’s dual. Set $\Lambda=G\mathbb{Z}^d$. Then $F\mathbb{Z}^d\subseteq\Lambda^\ast$. Given positive $\lambda_1,\dots,\lambda_d$, let $\lambda$ be defined by (43). Set $\eta_i=\lambda/\lambda_i$, $i=1,\dots,d$. Consider the parallelepiped $\mathcal{P}$ defined by (44). Then
$$
\begin{equation}
\frac1{\eta_i}\prod_{j=1}^d\eta_j= \frac{\lambda_i\lambda^{d-1}}{\prod_{j=1}^d\lambda_j}=\lambda_i,\qquad i=1,\dots,d,
\end{equation}
\tag{45}
$$
that is,
$$
\begin{equation*}
\mathcal{P}^\ast=\bigl\{\mathbf z=(z_1,\dots,z_d)\in\mathbb{R}^d \bigm| |z_i|\leqslant\lambda_i,\ i=1,\dots,d \bigr\}.
\end{equation*}
\notag
$$
Thus, Theorem 21 actually states that there is a non-zero point of the unimodular lattice $\Lambda$ in $(d-1)\mathcal{P}$, provided that $\mathcal{P}^\ast$ contains a non-zero point of some sublattice of $\Lambda^\ast$. It is clear that in this statement the words ‘of some sublattice’ can be omitted. We obtain the following reformulation of Theorem 21. Theorem 22. Let $\Lambda$ be a full-rank lattice in $\mathbb{R}^d$ with determinant $1$. Let $\mathcal{P}$ be a parallelepiped in $\mathbb{R}^d$ centred at the origin with faces parallel to the coordinate planes. Then Note that, since $(\Lambda^\ast)^\ast=\Lambda$, we can interchange $\Lambda$ and $\Lambda^\ast$ in Theorem 22. In this form, Mahler’s theorem admits a rather transparent purely geometric proof. It can be described as follows. Suppose $\mathcal{P}^\ast$ contains a non-zero point $\mathbf{v}$ of $\Lambda^\ast$. We can assume that $\mathbf{v}$ is primitive. Consider $(\mathbb{R}\mathbf{v})^\perp$, the orthogonal complement to the one-dimensional subspace spanned by $\mathbf{v}$ and the cross-section $\mathcal{S}=\mathcal{P}\cap(\mathbb{R}\mathbf{v})^\perp$ (see Fig. 7). The set $\Gamma=\Lambda\cap(\mathbb{R}\mathbf{v})^\perp$ is a lattice of rank $d-1$ with determinant equal to $|\mathbf{v}|_2$, where $|\,{\cdot}\,|_2$ denotes the Euclidean norm. Hence, if a constant $c$ is chosen so that the $(d-1)$-dimensional volume of $c\mathcal{S}$ is smaller than $2^{d-1}|\mathbf{v}|_2$, then by Minkowski’s convex body theorem there is a non-zero point of $\Gamma$ in $c\mathcal{S}$, and therefore there is a non-zero point of $\Lambda$ in $c\mathcal{P}$. Finding an appropriate constant $c$ can be arranged with the help of Vaaler’s theorem [60] on central cross-sections of a cube. Consider the cube $\mathcal{B}=[-1,1]^d$. By Vaaler’s theorem the volume of any $(d-1)$-dimensional central cross-section of $\mathcal{B}$ is at least $2^{d-1}$. Consider the diagonal operators
$$
\begin{equation*}
A=\operatorname{diag}(\eta_1^{-1},\dots,\eta_d^{-1})\quad\text{and}\quad C=\operatorname{diag}(\lambda_1^{-1},\dots,\lambda_d^{-1}).
\end{equation*}
\notag
$$
Then By (45) $C$ coincides with the cofactor matrix of $A$, that is, $C=(\det A)(A^\ast)^{-1}$. Therefore, since $\mathcal{S}\perp\mathbf{v}$, we have
$$
\begin{equation*}
\frac{\operatorname{vol}\mathcal{S}}{|\mathbf v|_2}= (\det A)^{-1}\frac{\operatorname{vol}(A\mathcal{S})} {|(A^\ast)^{-1}\mathbf v|_2}= \frac{\operatorname{vol}(A\mathcal{S})}{|C\mathbf v|_2}\geqslant \frac{2^{d-1}}{\sqrt d}\,.
\end{equation*}
\notag
$$
Thus, by choosing $c=\bigl(\sqrt d\,\bigr)^{1/(d-1)}$ we obtain $\operatorname{vol}(c\mathcal{S})\geqslant2^{d-1}|\mathbf{v}|_2$. As we said before, the rest is done by applying Minkowski’s convex body theorem to $c\mathcal{S}$ and $\Gamma$. The argument we have just discussed actually proves a stronger statement than Theorem 22. It proves that
$$
\begin{equation}
\mathcal{P}^\ast\cap\Lambda^\ast\ne \{\mathbf 0\}\ \ \implies\ \ c\mathcal{P}\cap\Lambda\ne \{\mathbf 0\}
\end{equation}
\tag{46}
$$
for $c=\bigl(\sqrt d\,\bigr)^{1/(d-1)}$, a constant less than ${d-1}$, which, moreover, tends to $1$ as $d\to\infty$. However, it is worth mentioning that a combination of Mahler’s theorem on successive minima proved in [61] with Minkowski’s theorem on successive minima (both theorems can be found in Schmidt’s book [11]) provides an improvement of Theorem 21 with constant $c^2$. A detailed account can be found in [62]. In the same paper some further improvements of Theorem 21 are presented. 3.4.3. Two two-parametric families of parallelepipedsLet us address Theorem 11 of Khintchine once again, this time in the light of Theorem 22. Consider the lattice
$$
\begin{equation*}
\Lambda=\begin{pmatrix} 1 &0 &0 & \dots &0 \\ -\theta_1 &1 &0 & \dots &0 \\ -\theta_2 &0 &1 & \dots &0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ -\theta_n &0 &0 & \dots &1 \end{pmatrix}\mathbb{Z}^{n+1}.
\end{equation*}
\notag
$$
Then the dual lattice is
$$
\begin{equation*}
\Lambda^\ast=\begin{pmatrix} 1 & \theta_1 & \theta_2 & \dots & \theta_n \\ 0 &1 &0 & \dots &0 \\ 0 &0 &1 & \dots &0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 &0 &0 & \dots &1 \end{pmatrix}\mathbb{Z}^{n+1}.
\end{equation*}
\notag
$$
For all positive $t$, $\gamma$, $s$, and $\delta$ define the parallelepipeds
$$
\begin{equation}
\mathcal{P}(t,\gamma)=\biggl\{\mathbf z=(z_1,\dots,z_{n+1}) \in\mathbb{R}^{n+1} \biggm| \begin{aligned} \, &|z_1|\leqslant t, \\ &|z_i|\leqslant t^{-\gamma},\quad i=2,\dots,n+1 \end{aligned} \biggr\}
\end{equation}
\tag{47}
$$
and
$$
\begin{equation}
\mathcal{Q}(s,\delta)=\biggl\{\mathbf z=(z_1,\dots,z_{n+1})\in \mathbb{R}^{n+1} \biggm| \begin{aligned} \, &|z_1|\leqslant s^{-\delta}, \\ &|z_i|\leqslant s,\quad i=2,\dots,n+1 \end{aligned} \biggr\}.
\end{equation}
\tag{48}
$$
Then
$$
\begin{equation}
\begin{aligned} \, \omega(\Theta)&=\sup\Bigl\{\gamma\geqslant\frac{1}{n} \Bigm| \forall\,t_0\in\mathbb{R}\,\ \exists\,t>t_0\colon \mathcal{P}(t,\gamma)\cap\Lambda\ne \{\mathbf 0\}\Bigr\}, \\ \omega(L_\Theta)&=\sup\{\delta\geqslant n\mid \forall\,s_0\in\mathbb{R}\,\ \exists\,s>s_0\colon\mathcal{Q}(s,\delta)\cap \Lambda^\ast\ne \{\mathbf 0\}\}. \end{aligned}
\end{equation}
\tag{49}
$$
Each parallelepiped in (48) is the pseudocompound of some parallelepiped in (47), and vice versa. Indeed, if
$$
\begin{equation}
t=s^{((n-1)\delta+n)/n}\quad\text{and}\quad \gamma=\frac{\delta}{(n-1)\delta+n}\,,
\end{equation}
\tag{50}
$$
then $\mathcal{Q}(s,\delta)=\mathcal{P}(t,\gamma)^\ast$. Conversely, if then $\mathcal{P}(t,\gamma)=\mathcal{Q}(s,\delta)^\ast$. Let us apply Mahler’s theorem in disguise of Theorem 22. Assuming that (50) holds, we have
$$
\begin{equation*}
\mathcal{Q}(s,\delta)\cap\Lambda^\ast\ne \{\mathbf 0\}\ \ \implies\ \ (d-1)\mathcal{P}(t,\gamma)\cap\Lambda\ne \{\mathbf 0\},
\end{equation*}
\notag
$$
so that, in view of (49),
$$
\begin{equation*}
\omega(L_\Theta)\geqslant\delta\ \ \implies\ \ \omega(\Theta)\geqslant\gamma=\frac{\delta}{(n-1)\delta+n}\,.
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation}
\omega(\Theta)\geqslant\frac{\omega(L_\Theta)}{(n-1)\omega(L_\Theta)+n}\,.
\end{equation}
\tag{52}
$$
Assuming that (51) holds, we have
$$
\begin{equation*}
\mathcal{P}(t,\gamma)\cap\Lambda\ne \{\mathbf 0\}\ \ \implies\ \ (d-1)\mathcal{Q}(s,\delta)\cap\Lambda^\ast\ne \{\mathbf 0\},
\end{equation*}
\notag
$$
so that, in view of (49) again,
$$
\begin{equation*}
\omega(\Theta)\geqslant\gamma\ \ \implies\ \ \omega(L_\Theta)\geqslant\delta=n\gamma+n-1.
\end{equation*}
\notag
$$
Thus, It can easily be verified that (52) and (53) are just the right- and left-hand inequalities in (18), respectively. $\Box$ 3.4.4. Uniform exponents and an analogue of Mahler’s theoremThe proof of Theorem 22 presented in § 3.4.2 is based on Minkowski’s convex body theorem, Vaaler’s theorem on central cross-sections of a cube, and the fact that for any primitive vector $\mathbf{v}$ of a unimodular lattice $\Lambda$ the set $\Lambda^\ast\cap(\mathbb{R}\mathbf{v})^\perp$ is a lattice of rank ${d-1}$ with determinant equal to the Euclidean norm of $\mathbf{v}$. This fact can be reformulated as the equality of the determinants of the lattices $\Lambda\cap(\mathbb{R}\mathbf{v})$ and $\Lambda^\ast\cap(\mathbb{R}\mathbf{v})^\perp$ of ranks $1$ and ${d-1}$, respectively. A more general statement holds. It is very useful when working with uniform exponents. Proposition 5. Let $\Lambda$ be a full-rank lattice in $\mathbb{R}^d$, and let $\det\Lambda=1$. Let $\mathcal{L}$ be a $k$-dimensional subspace of $\mathbb{R}^d$ and let the lattice $\Gamma=\mathcal{L}\cap\Lambda$ have rank $k$. Consider the orthogonal complement $\mathcal{L}^\perp$ and set $\Gamma^\perp=\mathcal{L}^\perp\cap\Lambda^\ast$. Then $\Gamma^\perp$ is a lattice of rank ${d-k}$ and This statement is rather well known; its proof can be found, for instance, in [63] or [36]. The nature of uniform exponents requires working with sublattices of rank $2$, so that Proposition 5 is used for ${k=2}$. Correspondingly, when working with two-dimensional and $(d-2)$-dimensional subspaces, it is natural to involve $(d-2)$nd pseudocompounds instead of $(d-1)$st ones. Let $\mathbf{e}_1,\dots,\mathbf{e}_d$ be the standard basis of $\mathbb{R}^d$. For each multivector $\mathbf{Z}\in\bigwedge^2\mathbb{R}^d$ consider its representation
$$
\begin{equation*}
\mathbf Z=\sum_{1\leqslant i<j\leqslant d}Z_{ij}\, \mathbf e_i\wedge\mathbf e_j.
\end{equation*}
\notag
$$
Definition 16. Let $\eta_1,\dots,\eta_d$ be positive real numbers. Consider the parallelepiped Then the parallelepiped is called the $(d-2)$nd pseudocompound of $\mathcal{P}$. Given a full-rank lattice $\Lambda$ in $\mathbb{R}^d$ and its dual lattice $\Lambda^\ast$, we denote by $\Lambda^{\circledast}$ the set of all the decomposable elements of the lattice $\bigwedge^2\Lambda^\ast$, that is,
$$
\begin{equation*}
\Lambda^{\circledast}=\{\mathbf z_1\wedge\mathbf z_2\mid \mathbf z_1,\mathbf z_2\in\Lambda^\ast\}.
\end{equation*}
\notag
$$
The following theorem proved in [64] is an analogue of Theorem 22. Theorem 23. Let $\Lambda$ be a full-rank lattice in $\mathbb{R}^d$ with determinant $1$. Let $\mathcal{P}$ be a parallelepiped in $\mathbb{R}^d$ centred at the origin with faces parallel to coordinate planes. Then where The proof of Theorem 23 resembles pretty much the proof of Theorem 22 described in § 3.4.2. It is also based on three facts. The first is the same Minkowski convex body theorem. The second is Vaaler’s theorem on central cross-sections of a cube: this time it is applied to $(d-2)$-dimensional sections. The third ingredient is Proposition 5 for $k=2$. Theorem 23 is the key tool for proving Theorem 13 on uniform exponents. It is applied within the framework of a parametric construction, which is most comprehensively described in terms of ‘nodes’ and ‘leaves’. 3.4.5. Families of ‘nodes’, ‘antinodes’, and ‘leaves’Let us describe the construction of ‘nodes’ and ‘leaves’ in the form adapted for the proof of the right-hand inequality in (22), that is, the inequality
$$
\begin{equation}
\widehat\omega(\Theta)\geqslant \frac{1-\widehat\omega(L_\Theta)^{-1}}{n-1}\,.
\end{equation}
\tag{54}
$$
We adopt the notation (47) and (48) once again. Fix arbitrary $h,\alpha,\beta\in\mathbb{R}$ such that
$$
\begin{equation*}
h>1,\quad \beta>0, \quad\text{and}\quad 0<\alpha\leqslant\beta,
\end{equation*}
\notag
$$
and set To each $r$ in the interval $h\leqslant r\leqslant H$ we assign the parallelepiped
$$
\begin{equation}
\mathcal{Q}_r=\biggl\{\mathbf z=(z_1,\dots,z_d) \in\mathbb{R}^d \biggm| \begin{aligned} \, &|z_1|\leqslant (hH/r)^{-\alpha} \\ &|z_i|\leqslant r,\ i=2,\dots,n+1 \end{aligned} \biggr\}.
\end{equation}
\tag{55}
$$
It is easy to see that $\mathcal{Q}_r$ belongs to (48):
$$
\begin{equation*}
\mathcal{Q}_r=\mathcal{Q}\biggl(r,\alpha\log_r\frac{hH}{r}\biggr).
\end{equation*}
\notag
$$
Consider the following three families of parallelepipeds:
$$
\begin{equation*}
\begin{aligned} \, \mathfrak S&=\mathfrak S(h,\alpha,\beta)= \{\mathcal{Q}_r\mid h\leqslant r\leqslant\sqrt{hH}\}, \\ \mathfrak A&=\mathfrak A(h,\alpha,\beta)= \{\mathcal{Q}_r\mid \sqrt{hH}\leqslant r\leqslant H\}, \\ \mathfrak L&=\mathfrak L(h,\alpha,\beta)= \{\mathcal{Q}(r,\alpha) \mid h\leqslant r\leqslant H\}. \end{aligned}
\end{equation*}
\notag
$$
We call $\mathfrak S$ the ‘stem’ family, $\mathfrak A$ the ‘anti-stem’ family, and $\mathfrak L$ the ‘leaves’ family. We call each element of $\mathfrak S$ a ‘node’, each element of $\mathfrak A$ an ‘anti-node’, and each element of $\mathfrak L$ a ‘leaf’. We say that a ‘leaf’ $\mathcal{Q}(r,\alpha)$ is produced by a ‘node’ or an ‘anti-node’ $\mathcal{Q}_{r'}$ if We call $\mathcal{Q}_h$ the root ‘node’. ‘Nodes’ and ‘leaves’ enjoy the following properties:
These properties are illustrated by Fig. 8, where we use $u$ and $v$ to denote $\max(|z_2|,\dots,|z_{n+1}|)$ and $|z_1|$, respectively. The construction we have described is adapted, as we said before, for the proof of (54), which is the right-hand inequality in (22). Nevertheless, Fig. 8 can be used unaltered for the proof of the left-hand inequality in (22). Moreover, it can also be used in more general problems: in the problem of approximating zero by values of several linear forms (§ 4) and in the analogous problem with weights (§ 6). 3.4.6. An outline of the proof of the transference inequalities for uniform exponentsLet us demonstrate how to apply Theorem 23 and the parametric construction described above to prove the right-hand inequality in (22). By analogy with (49), we have
$$
\begin{equation}
\begin{aligned} \, \widehat\omega(\Theta) & =\sup\biggl\{ \gamma\geqslant\frac{1}{n} \Bigm| \exists\,t_0\in\mathbb{R}\colon \forall\,t>t_0\ \ \mathcal{P}(t,\gamma)\cap \Lambda\ne \{\mathbf 0\}\biggr\}, \\ \widehat\omega(L_\Theta) & =\sup\bigl\{\delta\geqslant n\mid \exists\,s_0\in\mathbb{R}\colon \forall\,s>s_0\ \ \mathcal{Q}(s,\delta)\cap \Lambda^\ast\ne \{\mathbf 0\}\}. \end{aligned}
\end{equation}
\tag{56}
$$
Fix $s>1$ and $\delta\geqslant n$. Like in (50), set
$$
\begin{equation*}
t=s^{((n-1)\delta+n)/n}\quad\text{and}\quad \gamma=\frac{\delta}{(n-1)\delta+n}\,.
\end{equation*}
\notag
$$
Also set
$$
\begin{equation*}
h=s,\qquad \beta=\delta\quad\text{and}\quad \alpha=\frac{(n-1)\delta+n}{n}\,.
\end{equation*}
\notag
$$
Note that, for such a choice of parameters, the quantities $\gamma$ and $\alpha$ are related by Then $\mathcal{Q}(s,\delta)$ is the root ‘node’. If it contains a non-zero point of $\Lambda^\ast$, then by Theorem 22 or, to be more exact, by its improved version (46) we have
$$
\begin{equation}
\mathcal{Q}(s,\delta)\cap\Lambda^\ast\ne \{\mathbf 0\}\ \ \implies\ \ c_1\mathcal{P}(t,\gamma)\cap\Lambda\ne \{\mathbf 0\},
\end{equation}
\tag{58}
$$
where $c_1=d^{1/(2(d-1))}$. If $\mathcal{Q}(s,\delta)$ does not contain non-zero points of $\Lambda^\ast$, then we assume that every ‘leaf’ $\mathcal{Q}(r,\alpha)$ in $\mathfrak L$ does contain non-zero points of $\Lambda^\ast$ and consider a ‘leaf’ $\mathcal{Q}(r_0,\alpha)$, whose existence is guaranteed by property (labv) in the previous section. Then there exist non-collinear points $\mathbf{v}_1$ and $\mathbf{v}_2$ in $\Lambda^\ast$ such that
$$
\begin{equation*}
\mathbf v_1\in\mathcal{Q}_{r_0} \quad\text{and}\quad \mathbf v_2\in\mathcal{Q}_{hH/r_0}.
\end{equation*}
\notag
$$
The parameters of the parallelepipeds $\mathcal{Q}_{r_0}$ and $\mathcal{Q}_{hH/r_0}$ are known, hence we can estimate the coordinates of the multivector
$$
\begin{equation*}
\mathbf v_1\wedge\mathbf v_2= \sum_{1\leqslant i<j\leqslant n+1}V_{ij}\mathbf e_i\wedge\mathbf e_j
\end{equation*}
\notag
$$
from above. Detailed calculations can be found in [64]. The estimates lead eventually to the key relation
$$
\begin{equation*}
\mathbf v_1\wedge\mathbf v_2\in2\mathcal{P}(t,\gamma)^{\circledast},
\end{equation*}
\notag
$$
which enables us to apply Theorem 23. We obtain the following chain of implications:
$$
\begin{equation}
\begin{aligned} \, \nonumber &\mathcal{Q}(r,\alpha)\cap\Lambda^\ast\ne \{\mathbf 0\}\text{ for each } r\in[h,H] \\ &\qquad\implies\ \ 2\mathcal{P}(t,\gamma)^{\circledast}\cap \Lambda^{\circledast}\ne \{\mathbf 0\}\ \ \implies\ \ c_2\mathcal{P}(t,\gamma)\cap\Lambda\ne \{\mathbf 0\}, \end{aligned}
\end{equation}
\tag{59}
$$
where $c_2=\bigl(2d(d-1)\bigr)^{1/(2(d-2))}$. Thus, if $\mathcal{Q}(s,\delta)$ contains a non-zero point of $\Lambda^\ast$, then by (58) there is a non-zero point of $\Lambda$ in $c_1\mathcal{P}(t,\gamma)$. If $\mathcal{Q}(s,\delta)$ does not contain non-zero points of $\Lambda^\ast$, but each $\mathcal{Q}(r,\alpha)$ in $\mathfrak L$ does, then by (59) a non-zero point of $\Lambda$ can be found in $c_2\mathcal{P}(t,\gamma)$. Taking (57) into account we obtain
$$
\begin{equation*}
\widehat\omega(L_\Theta)\geqslant\alpha\ \ \implies\ \ \widehat\omega(\Theta)\geqslant\gamma=\frac{1-\alpha^{-1}}{n-1}\,;
\end{equation*}
\notag
$$
hence
$$
\begin{equation*}
\widehat\omega(\Theta)\geqslant\frac{1-\widehat\omega(L_\Theta)^{-1}}{n-1}\,.
\end{equation*}
\notag
$$
This is how the right-hand inequality in (22) is proved. Proving the left-hand inequality in (22) requires a ‘converse’ argument. Instead of $\mathcal{Q}_r$, the parallelepipeds
$$
\begin{equation*}
\mathcal{P}_r=\biggl\{\,\mathbf z=(z_1,\dots,z_d) \in\mathbb{R}^d \biggm| \begin{aligned} \, &|z_1|\leqslant r \\ &|z_i|\leqslant (hH/r)^{-\alpha},\ \ i=2,\dots,n+1 \end{aligned} \biggr\}
\end{equation*}
\notag
$$
are to be considered, $\mathcal{Q}$ should be replaced by $\mathcal{P}$ in the definitions of the families $\mathfrak S$, $\mathfrak A$, and $\mathfrak L$, and, moreover, $u$ and $v$ in Fig. 8 should denote $|z_1|$ and $\max(|z_2|,\dots,|z_{n+1}|)$ respectively (in the previous section it was the other way around). Note that nothing changes in Fig. 8 itself. Upon fixing $t>1$ and $\gamma\geqslant1/n$, we should set, as in (51), and also
$$
\begin{equation*}
h=t,\qquad \beta=\gamma,\quad\text{and}\quad \alpha=\frac{n\gamma}{n\gamma+n-1}\,.
\end{equation*}
\notag
$$
For such a choice of parameters the quantities $\delta$ and $\alpha$ are related by Our further argument is absolutely analogous to the one discussed above. It leads to the implication
$$
\begin{equation*}
\widehat\omega(\Theta)\geqslant\alpha\ \ \implies\ \ \widehat\omega(L_\Theta)\geqslant\delta=\frac{n-1}{1-\alpha}\,,
\end{equation*}
\notag
$$
from which we obtain
$$
\begin{equation*}
\widehat\omega(L_\Theta)\geqslant \frac{n-1}{1-\widehat\omega(\Theta)}\,.
\end{equation*}
\notag
$$
This is how the left-hand inequality in (22), and therefore the whole of Theorem 13, is proved. $\Box$ 3.4.7. The empty cylinder lemma and ‘mixed’ inequalitiesInequalities (22) estimate uniform exponents from below. For estimating them from above — for instance, as in Theorem 15 of Schmidt and Summerer — the following rather simple statement proved in [46] is very efficient. It is reasonable to call it the ‘empty cylinder lemma’. As before, let us denote by $\mathcal{L}=\mathcal{L}(\Theta)$ the one-dimensional subspace spanned by $(1,\theta_1,\dots,\theta_n)$ and by $\mathcal{L}^\perp$ its orthogonal complement. For $\mathbf{u}\in\mathbb{R}^{n+1}$ let us denote by $r(\mathbf{u})$ the Euclidean distance from $\mathbf{u}$ to $\mathcal{L}$ and by $h(\mathbf{u})$ the Euclidean distance from $\mathbf{u}$ to $\mathcal{L}^\perp$. Lemma 2 (empty cylinder lemma). Let $t$, $\alpha$, and $\beta$ be positive real numbers such that $t^{\beta-\alpha}\geqslant2$ (or, equivalently, $t^{-\alpha}-t^{-\beta}\geqslant t^{-\beta}$). Suppose $\mathbf{v}\in\mathbb{Z}^{n+1}$ satisfies Consider the half-open cylinder Then $\mathcal{C}\cap\mathbb{Z}^{n+1}=\varnothing$. The geometric meaning of this statement is that the cylinder $\mathcal{C}$ is contained in the open ‘layer’ between the planes determined by the equations $\langle\mathbf{v},\mathbf{u}\rangle=0$ and $\langle\mathbf{v},\mathbf{u}\rangle=1$, where $\langle\,\cdot\,{,}\,\cdot\,\rangle$ denotes, as before, the inner product in $\mathbb{R}^{n+1}$. To see this, consider the two-dimensional subspace $\pi$ spanned by $\mathcal{L}$ and $\mathbf{v}$. It is shown in Fig. 9. Consider an arbitrary point $\mathbf{w}\in\mathcal{C}$, and let $\mathbf{w}'$ be its orthogonal projection onto $\pi$. Within the plane $\pi$ the functionals $h(\,\cdot\,)$ and $r(\,\cdot\,)$ can be identified with the absolute values of coordinates of points with respect to the coordinate axes $\mathcal{L}$ and $\mathcal{L}^\perp\cap\pi$. Then
$$
\begin{equation*}
\langle\mathbf v,\mathbf w\rangle= \langle\mathbf v,\mathbf w'\rangle> \begin{pmatrix} t^{\alpha-1-\beta} & t^\alpha \end{pmatrix}\begin{pmatrix} -t \\ t^{-\beta} \end{pmatrix}=0
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\langle\mathbf v,\mathbf w\rangle= \langle\mathbf v,\mathbf w'\rangle< \begin{pmatrix} t^{\alpha-1-\beta} & t^\alpha \end{pmatrix}\begin{pmatrix} t \\ t^{-\alpha}-t^{-\beta} \end{pmatrix}=1.
\end{equation*}
\notag
$$
Thus, $0<\langle\mathbf{v},\mathbf{w}\rangle<1$, and therefore $\mathbf{w}$ cannot be an integer point. Lemma 2 is proved. Let us outline a proof of Theorem 15, due to Schmidt and Summerer, which involves Lemma 2. We leave out some details for simplicity. They can be found in [46]. Consider an arbitrary non-zero point $\mathbf{v}\in\mathbb{Z}^{n+1}$, and set $t=t(\mathbf{v})$ to be the smallest positive real number such that the cylinder
$$
\begin{equation}
\mathcal{C}_{\mathbf v}=\biggl\{\mathbf u\in\mathbb{R}^{n+1}\Bigm| r(\mathbf u)\leqslant t,\ h(\mathbf u)\leqslant t\, \frac{r(\mathbf v)}{h(\mathbf v)}\biggr\}
\end{equation}
\tag{62}
$$
contains a non-zero point of $\mathbb{Z}^{n+1}$. The geometric meaning of the inequalities determining $\mathcal{C}_{\mathbf{v}}$ is that each of the two bases of this cylinder has exactly one common point with the plane $\langle\mathbf{v},\mathbf{u}\rangle=0$. Also define $\gamma=\gamma(\mathbf{v})$, $\alpha=\alpha(\mathbf{v})$, and $\beta=\beta(\mathbf{v})$ by
$$
\begin{equation}
r(\mathbf v)=h(\mathbf v)^{-\gamma},\qquad h(\mathbf v)=t^\alpha,\quad\text{and}\quad \alpha=\frac{1+\beta}{1+\gamma}\,.
\end{equation}
\tag{63}
$$
It can easily be verified that for such a choice of parameters $\mathbf{v}$ satisfies (60) and $\mathcal{C}_{\mathbf{v}}$ satisfies
$$
\begin{equation*}
\mathcal{C}_{\mathbf v}=\{\mathbf u\in\mathbb{R}^{n+1}\mid r(\mathbf u)\leqslant t,\ h(\mathbf u)\leqslant t^{-\beta}\}.
\end{equation*}
\notag
$$
Hence, if the condition is fulfilled, then $\mathcal{C}_{\mathbf{v}}$ can be supplemented with the non-empty cylinders $\mathcal{C}$ and $-\mathcal{C}$ defined by (61), which do not contain integer points by Lemma 2 (see Fig. 10). Thus, to each point $\mathbf{v}$ satisfying (64) we have assigned the cylinder $\mathcal{C}_{\mathbf{v}}$, which contains integer points on its boundary, and the cylinder
$$
\begin{equation*}
\mathcal{C}'_{\mathbf v}=\mathcal{C}_{\mathbf v}\cup \mathcal{C}\cup(-\mathcal{C}),
\end{equation*}
\notag
$$
which does not contain non-zero integer points in its interior. Furthermore, the condition (64) is guaranteed to be fulfilled if $h(\mathbf{v}_k)r(\mathbf{v}_k)^n$ is small enough. The reason is that the lattice that is the orthogonal projection of $\mathbb{Z}^{n+1}\cap(\mathbb{R}\mathbf{v})^{\perp}$ onto $\mathcal{L}^\perp$ has determinant equal to $h(\mathbf{v}_k)$. It follows by Minkowski’s convex body theorem that $t^n\leqslant2^nh(\mathbf{v}_k)/B$, where $B$ equals the volume of the $n$-dimensional Euclidean ball of radius $1$. Thus, if $h(\mathbf{v}_k)r(\mathbf{v}_k)^n\leqslant4^{-n}B$, then $t^n\leqslant2^{-n}r(\mathbf{v}_k)^{-n}=(2t^{\alpha-1-\beta})^{-n}$, that is, $t^{\beta-\alpha}\geqslant2$, which is equivalent to (64). Geometrically, this means that the cylinder $\mathcal{C}_{\mathbf{v}}$ does not ‘reach’ the plane $\langle\mathbf{v},\mathbf{u}\rangle=1$. In order to prove the left-hand inequality in (25) consider a sequence of points $\mathbf{v}_k$ such that
$$
\begin{equation*}
h(\mathbf v_k)\to\infty \quad\text{and}\quad \gamma_k=\gamma(\mathbf v_k)\to\omega(\Theta) \quad\text{as}\ k\to\infty.
\end{equation*}
\notag
$$
If $\omega(\Theta)>1/n$, then the quantity $h(\mathbf{v}_k)r(\mathbf{v}_k)^n$ is small for large $k$, that is, condition (64) is fulfilled. Set $\alpha_k=\alpha(\mathbf{v})$ and $\beta_k=\beta(\mathbf{v})$. Then
$$
\begin{equation*}
\omega(L_\Theta)\geqslant\limsup_{k\to\infty}\beta_k \quad\text{and}\quad \widehat\omega(L_\Theta)\leqslant\liminf_{k\to\infty}\alpha_k.
\end{equation*}
\notag
$$
Taking (63) into account we obtain
$$
\begin{equation*}
\widehat\omega(L_\Theta)\leqslant \liminf_{k\to\infty}\frac{1+\beta_k}{1+\gamma_k}\leqslant \frac{1+\limsup_{k\to\infty}\beta_k}{1+\lim_{k\to\infty}\gamma_k}\leqslant \frac{1+\omega(L_\Theta)}{1+\omega(\Theta)}\,.
\end{equation*}
\notag
$$
The left-hand inequality in (25) is proved. To prove the right-hand inequality in (25), consider a sequence of points $\mathbf{v}_k$ such that
$$
\begin{equation*}
h(\mathbf v_k)\to0 \quad\text{and}\quad \gamma_k=\gamma(\mathbf v_k)\to\omega(L_\Theta)^{-1}\quad\text{as}\ k\to\infty.
\end{equation*}
\notag
$$
If $\omega(L_\Theta)>n$, then the quantity $h(\mathbf{v}_k)r(\mathbf{v}_k)^n$ is small for large $k$, that is, condition (64) is fulfilled again. Set $\alpha_k=\alpha(\mathbf{v})$ and $\beta_k=\beta(\mathbf{v})$. Then
$$
\begin{equation*}
\omega(\Theta)\geqslant\limsup_{k\to\infty}\beta_k^{-1}\quad\text{and} \quad \widehat\omega(\Theta)\leqslant\liminf_{k\to\infty}\alpha_k^{-1}.
\end{equation*}
\notag
$$
Taking (63) into account we obtain
$$
\begin{equation*}
\widehat\omega(\Theta)\leqslant \liminf_{k\to\infty}\frac{1+\gamma_k}{1+\beta_k}\leqslant \frac{1+\lim_{k\to\infty}\gamma_k}{1+\liminf_{k\to\infty}\beta_k}\leqslant \frac{1+\omega(L_\Theta)^{-1}}{1+\omega(\Theta)^{-1}}\,.
\end{equation*}
\notag
$$
The right-hand inequality in (25) is also proved. $\Box$ 3.4.8. Parametric geometry of numbersDeveloping the ideas proposed by Schmidt in [65], Schmidt and Summerer published a fundamental paper [66] in 2009, where they proposed a new point of view at the problems under discussion. They called their approach ‘parametric geometry of numbers’. Most generally parametric geometry of numbers can be outlined as follows. Let $\Lambda$ be a full-rank lattice with determinant $1$ in $\mathbb{R}^d$. Consider the cube $\mathcal{B}=[-1,1]^d$. Define the space of parameters by
$$
\begin{equation*}
\mathcal{T}=\{\boldsymbol{\tau}=(\tau_1,\dots,\tau_d)\in\mathbb{R}^d\mid \tau_1+\cdots+\tau_d=0\}.
\end{equation*}
\notag
$$
For each $\boldsymbol{\tau}\in\mathcal{T}$ set
$$
\begin{equation*}
\mathcal{B}_{\boldsymbol{\tau}}=\operatorname{diag}(e^{\tau_1},\dots, e^{\tau_d})\,\mathcal{B}.
\end{equation*}
\notag
$$
Let $\mu_k(\mathcal{B}_{\boldsymbol{\tau}},\Lambda)$, $k=1,\dots,d$, denote the $k$th successive minimum of $\mathcal{B}_{\boldsymbol{\tau}}$ with respect to $\Lambda$, that is, the smallest positive $\mu$ such that $\mu\mathcal{B}_{\boldsymbol{\tau}}$ contains at least $k$ linearly independent vectors of $\Lambda$. Finally, for each $k=1,\dots,d$ consider the functions
$$
\begin{equation*}
L_k(\boldsymbol{\tau})=L_k(\Lambda,\boldsymbol{\tau})= \log\bigl(\mu_k(\mathcal{B}_{\boldsymbol{\tau}},\Lambda)\bigr)\quad\text{and}\quad S_k(\boldsymbol{\tau})=S_k(\Lambda,\boldsymbol{\tau})= \sum_{1\leqslant j\leqslant k}L_j(\Lambda,\boldsymbol{\tau}).
\end{equation*}
\notag
$$
Many problems in Diophantine approximation can be formulated as questions concerning the asymptotic behaviour of $L_k(\boldsymbol{\tau})$ and $S_k(\boldsymbol{\tau})$. Each problem requires a proper choice of $\Lambda$ and a subset of $\mathcal{T}$ with respect to which the asymptotics of these functions are to be studied. In the case of the problems under discussion in this section $\boldsymbol{\tau}$ is supposed to tend to infinity along certain one-dimensional subspaces of $\mathcal{T}$. Let $\mathfrak{T}$ be the path in $\mathcal{T}$ determined by a mapping $s\mapsto\boldsymbol{\tau}(s)$, $s\in[0,\infty)$. For our purposes we may assume that this mapping is linear. Definition 17. Given a lattice $\Lambda$ and a path $\mathfrak{T}$, let $k\in\{1,\dots,d\}$. Then the quantities are called the $k$th lower and upper Schmidt–Summerer exponents of the first type, respectively. Definition 18. Given a lattice $\Lambda$ and a path $\mathfrak{T}$, let $k\in\{1,\dots,d\}$. Then the quantities are called the $k$th lower and upper Schmidt–Summerer exponents of the second type, respectively. For the problem of simultaneous approximation and the problem of approximating zero with the values of a linear form, the lattices and paths can be chosen as follows. Set $d=n+1$. As in § 3.4.3, consider the lattices
$$
\begin{equation}
\Lambda=\begin{pmatrix} 1 &0 &0 & \dots &0 \\ -\theta_1 &1 &0 & \dots &0 \\ -\theta_2 &0 &1 & \dots &0 \\ \phantom{-} \vdots & \vdots & \vdots & \ddots & \vdots \\ -\theta_n &0 &0 & \dots &1 \end{pmatrix}\mathbb{Z}^{n+1}\quad\text{and}\quad \Lambda^\ast=\begin{pmatrix} 1 & \theta_1 & \theta_2 & \dots & \theta_n \\ 0 &1 &0 & \dots &0 \\ 0 &0 &1 & \dots &0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 &0 &0 & \dots &1 \end{pmatrix}\mathbb{Z}^{n+1}.
\end{equation}
\tag{65}
$$
We define the paths $\mathfrak{T}$ and $\mathfrak{T}^\ast$ by the mappings
$$
\begin{equation}
\begin{gathered} \, s\mapsto\boldsymbol{\tau}(s)=\bigl(\tau_1(s),\dots,\tau_{n+1}(s)\bigr), \\ \tau_1(s)=s,\quad \tau_2(s)=\cdots=\tau_{n+1}(s)=-\frac{s}{n}\,, \end{gathered}
\end{equation}
\tag{66}
$$
and
$$
\begin{equation*}
\begin{gathered} \, s\mapsto\boldsymbol{\tau}^\ast(s)= \bigl(\tau^\ast_1(s),\dots,\tau^\ast_{n+1}(s)\bigr), \\ \tau^\ast_1(s)=-ns,\quad \tau^\ast_2(s)=\cdots=\tau^\ast_{n+1}(s)=s, \end{gathered}
\end{equation*}
\notag
$$
respectively. Then $\boldsymbol{\tau}^\ast(s)=-n\boldsymbol{\tau}(s)$, that is, $\mathfrak{T}\cup\mathfrak{T}^\ast$ is a one-dimensional subspace of $\mathcal{T}$. As we noted in § 3.4.3, $\Lambda^\ast$ is the dual of $\Lambda$. Regular and uniform Diophantine exponents are related to Schmidt–Summerer exponents by the equalities
$$
\begin{equation}
\begin{gathered} \, \bigl(1+\omega(\Theta)\bigr)\bigl(1+\underline{\varphi}_1(\Lambda,\mathfrak{T})\bigr)= \bigl(1+\widehat\omega(\Theta)\bigr) \bigl(1+\overline{\varphi}_1(\Lambda,\mathfrak{T})\bigr)=\frac{n+1}{n}\,, \\ \bigl(1+\omega(L_\Theta)\bigr) \bigl(1+\underline{\varphi}_1(\Lambda^\ast,\mathfrak{T}^\ast)\bigr)= \bigl(1+\widehat\omega(L_\Theta)\bigr) \bigl(1+\overline{\varphi}_1(\Lambda^\ast,\mathfrak{T}^\ast)\bigr)=n+1. \end{gathered}
\end{equation}
\tag{67}
$$
They are deduced directly from the definitions (see [66] and [37]). We note that the analogous relations for Schmidt–Summerer exponents with indices greater than one hold in the context of problems of approximating a given subspace by rational subspaces of fixed dimension. In those problems so-called intermediate exponents arise. They were investigated in [67], [40], [41], and [37]; the latter paper contains equalities connecting Schmidt–Summerer exponents and intermediate exponents. By (67) each relation proved for Schmidt–Summerer exponents generates a relation for regular and uniform Diophantine exponents. As for relations for Schmidt–Summerer exponents, they are obtained by analysing the properties of the functions $L_k(\boldsymbol{\tau})$ and $S_k(\boldsymbol{\tau})$. The first observation that can be made is that $L_k(\boldsymbol{\tau})$ and $S_k(\boldsymbol{\tau})$ are continuous and piecewise linear on $\mathcal{T}$ and their restrictions to the paths $\mathfrak{T}$ and $\mathfrak{T}^\ast$ have two possible slopes on intervals of linearity. Next, a very important role in the analysis of the behaviour of $L_k(\boldsymbol{\tau})$ and $S_k(\boldsymbol{\tau})$ is played by two classical theorems on successive minima, a theorem of Minkowski [9] and a theorem of Mahler [61]. Let us state these theorems for parallelepipeds $\mathcal{B}_{\boldsymbol{\tau}}$ and lattices $\Lambda$ and $\Lambda^\ast$ (assuming that $d=n+1$). Theorem 25 (Mahler, 1938). For each $k=1,\dots,d$ and each $\boldsymbol{\tau}\in\mathcal{T}$ we have The following local properties, holding for every $\boldsymbol{\tau}\in\mathcal{T}$, are easily deduced from Theorems 24 and 25 (see [68]):
$$
\begin{equation*}
\begin{alignedat}{2} &{\rm(i)}&&\quad L_k(\Lambda,\boldsymbol{\tau})= -L_{d+1-k}(\Lambda^\ast,-\boldsymbol{\tau})+O(1),\quad k=1,\dots,d; \\ &{\rm(ii)}&&\quad S_k(\Lambda,\boldsymbol{\tau})= S_{d-k}(\Lambda^\ast,-\boldsymbol{\tau})+O(1),\quad k=1,\dots,d-1; \\ &{\rm(iii)}&&\quad S_1(\Lambda,\boldsymbol{\tau})\leqslant\cdots\leqslant \dfrac{S_k(\Lambda,\boldsymbol{\tau})}{k}\leqslant\cdots\leqslant \dfrac{S_{d-1}(\Lambda,\boldsymbol{\tau})}{d-1}\leqslant \dfrac{S_{d}(\Lambda,\boldsymbol{\tau})}{d}=O(1); \\ &{\rm(iv)}&&\quad \dfrac{S_1(\Lambda,\boldsymbol{\tau})}{d-1}\geqslant\cdots\geqslant \dfrac{S_k(\Lambda,\boldsymbol{\tau})}{d-k}\geqslant\cdots\geqslant S_{d-1}(\Lambda,\boldsymbol{\tau}). \end{alignedat}
\end{equation*}
\notag
$$
The constants implied by $O(\,\cdot\,)$ depend only on $d$. These properties, as well as Theorems 24 and 25, are valid for every unimodular lattice. In particular, they remain valid if $\Lambda$ is replaced by $\Lambda^\ast$. Properties (ii) and (iii) immediately imply the inequality
$$
\begin{equation}
S_1(\Lambda,\boldsymbol{\tau})\leqslant \frac{S_1(\Lambda^\ast,-\boldsymbol{\tau})}{d-1}+O(1),
\end{equation}
\tag{70}
$$
which (by applying (70) to $\Lambda$ and $\Lambda^\ast$) yields the following inequalities for Schmidt–Summerer exponents corresponding to the problem of simultaneous approximation and the problem of approximating zero with the values of a linear form:
$$
\begin{equation}
\underline{\varphi}_1(\Lambda^\ast,\mathfrak{T}^\ast)\leqslant \underline{\varphi}_1(\Lambda,\mathfrak{T})\leqslant \frac{\underline{\varphi}_1(\Lambda^\ast,\mathfrak{T}^\ast)}{n^2}\,.
\end{equation}
\tag{71}
$$
Rewriting (71) in terms of $\omega(\Theta)$ and $\omega(L_\Theta)$, with the help of (67) we obtain Khintchine’s inequalities (18). It was shown in [37] that this way to Khintchine’s inequalities enables one to improve them by splitting each of them into a chain of inequalities between the intermediate exponents mentioned above. The first authors to obtain such a chain of inequalities were Laurent and Bugeaud (see [40] and [41]). The restrictions of $L_1(\boldsymbol{\tau}),\dots,L_d(\boldsymbol{\tau})$ to the path $\mathfrak{T}$ enjoy a series of elementary properties: they are continuous, piecewise linear, they have two possible slopes on intervals of linearity, their values at each point are ordered as $L_1(\boldsymbol{\tau})\leqslant\cdots\leqslant L_d(\boldsymbol{\tau})$, and their sum $S_d(\boldsymbol{\tau})=L_1(\boldsymbol{\tau}) +\cdots+L_d(\boldsymbol{\tau})$ is bounded and non-positive. In all other respects the behaviour of $L_1,\dots,L_d$ is rather chaotic. Nevertheless, there is a class of $d$-tuples of functions with quite a regular behaviour which enjoy the above properties of $L_1,\dots,L_d$ and approximate $L_1,\dots,L_d$ up to bounded functions. This outstanding result is due to Roy [69]. Its formulation requires the concept of a $d$-system. We adapt the corresponding definition from [70] to the path $\mathfrak{T}$ defined by (66) (we recall that this path corresponds to the problem of simultaneous approximation). As before, we assume that $d=n+1$. Definition 19. Let $I\subset[0,\infty)$ be an interval with non-empty interior. A continuous piecewise linear map $\mathbf{P}=(P_1,\dots,P_d)\colon I\to\mathbb{R}^d$ is called a $d$-system on $I$ if the following conditions are fulfilled: Assuming that the path $\mathfrak{T}$ and the lattice $\Lambda=\Lambda(\Theta)$ are defined by (66) and (65), let us define a map $\mathbf{L}_\Theta\colon[0,\infty)\to\mathbb{R}^d$ by
$$
\begin{equation*}
s\mapsto\bigl(L_1\bigl(\Lambda,\boldsymbol{\tau}(s)\bigr),\dots, L_d\bigl(\Lambda,\boldsymbol{\tau}(s)\bigr)\bigr).
\end{equation*}
\notag
$$
Theorem 26 (Roy, 2015). For each non-zero $\Theta\in\mathbb{R}^n$ there exist $s_0\geqslant0$ and a $d$-system $\mathbf{P}$ on $[s_0,\infty)$ such that $\mathbf{L}_\Theta-\mathbf{P}$ is bounded on $[s_0,\infty)$. Conversely, for each $d$-system $\mathbf{P}$ on $[s_0,\infty)$ with arbitrary $s_0\geqslant0$ there exists a non-zero $\Theta\in\mathbb{R}^n$ such that $\mathbf{L}_\Theta-\mathbf{P}$ is bounded on $[s_0,\infty)$. Roy’s theorem proved to be much in demand for proving the existence of a point of $\Theta$ with prescribed Diophantine properties. Just this Theorem 26 enabled establishing the sharpness of many transference inequalities and also proving Theorem 19. 4. Several linear forms4.1. A general setting for the main problem of homogeneous linear Diophantine approximationGiven a matrix
$$
\begin{equation*}
\Theta=\begin{pmatrix} \theta_{11} & \dots & \theta_{1m} \\ \vdots & \ddots & \vdots \\ \theta_{n1} & \dots & \theta_{nm} \end{pmatrix},\qquad \theta_{ij}\in\mathbb{R},\quad m+n\geqslant3,
\end{equation*}
\notag
$$
consider the system of linear equations with variables $\mathbf{x}=(x_1,\dots,x_m)\in\mathbb{R}^m$ and $\mathbf{y}=(y_1,\dots,y_n)\in\mathbb{R}^n$. In its most general form, the main question of homogeneous linear Diophantine approximation is how small the quantity $|\Theta\mathbf{x}-\mathbf{y}|$ can be for $\mathbf{x}\in\mathbb{Z}^m$ and $\mathbf{y}\in \mathbb{Z}^n$ satisfying the restriction $0<|\mathbf{x}|\leqslant t$. In what follows, by $|\,{\cdot}\,|$ we denote the sup-norm. It is easy to see that for $m=1$ we obtain the problem of simultaneous approximation, and for $n=1$ we obtain the problem of approximating zero with the values of a linear form. Like in these problems, the corresponding Diophantine exponents are naturally defined. The following definition contains Definitions 8–11 as particular cases. Definition 20. The supremum of real $\gamma$ satisfying the condition that there exist arbitrarily large $t$ such that (respectively, for every $t$ large enough) the system admits a non-zero solution $(\mathbf{x},\mathbf{y}) \in\mathbb{Z}^m\oplus\mathbb{Z}^n$ is called the regular (respectively, uniform) Diophantine exponent of $\Theta$ and is denoted by $\omega(\Theta)$ (respectively, $\widehat\omega(\Theta)$). 4.2. Dirichlet’s theorem againIn the case of several linear forms an analogue of Theorems 1, 8, and 9 holds. It was also proved in Dirichlet’s paper [1], though, just as with these theorems, he formulated it originally in a weaker form. We recall that $|\,{\cdot}\,|$ denotes the sup-norm. Let us also denote by $\mathbb{R}^{n\times m}$ the set of $n\times m$ real matrices. Theorem 27 (Lejeune Dirichlet, 1842). Let $\Theta\in\mathbb{R}^{n\times m}$. Then, for each $t\geqslant1$, there is a non-zero pair $(\mathbf{x},\mathbf{y})\in\mathbb{Z}^m\oplus\mathbb{Z}^n$ such that We note again that Theorem 27 follows immediately from Minkowski’s convex body theorem in the form of Corollary 5 as applied to system (73). Similarly to (13), these inequalities are sharp. Moreover, Theorem 10 of Perron can be generalised to the matrix setting. Perron’s idea can be used to prove (for instance, this was done in Schmidt’s book [11]) that there exist matrices with algebraic entries that are badly approximable. Definition 21. A matrix $\Theta$ is called badly approximable if there is a positive constant $c$ depending only on $\Theta$ such that for every pair $(\mathbf{x},\mathbf{y})\in\mathbb{Z}^m\oplus\mathbb{Z}^n$ with non-zero $\mathbf{x}$ we have In the same book [11] by Schmidt it is proved that a matrix is badly approximable if and only if so is its transpose. In § 4.4.2 we show how to deduce this fact immediately from Mahler’s theorem. This fact is a manifestation of the transference principle, to which we already paid attention in § 3.3. 4.3. The transference theoremsIn § 3.3 we discussed the transference principle discovered by Khintchine, which relates the problem of simultaneous approximation and the problem of approximating zero by values of a linear form. This can be generalised to the case of an arbitrary matrix $\Theta\in\mathbb{R}^{n\times m}$. Let us denote the transposed matrix by $\Theta^\top$. 4.3.1. Inequalities for regular exponentsIn 1947 Dyson [71] generalised Khintchine’s transference principle (Theorem 11) as follows. Note that it follows from (75) and (74) that
$$
\begin{equation}
\omega(\Theta)=\frac{m}{n}\ \ \iff \ \ \omega(\Theta^\top)=\frac{n}{m}\,.
\end{equation}
\tag{76}
$$
A year later, Khintchine [35] published a simpler proof of Theorem 28. But actually, Theorem 28 could have been derived in 1937 as a corollary to Theorem 21 of Mahler. We discuss this in more detail in § 4.4. Note that any statement proved for an arbitrary matrix $\Theta\in\mathbb{R}^{n\times m}$ with arbitrary $n$ and $m$ gives rise automatically to the analogous statement for $\Theta^\top$. We simply need to swap the triple $(n,m,\Theta)$ for $(m,n,\Theta^\top)$. Therefore, for an arbitrary matrix $\Theta\in\mathbb{R}^{n\times m}$ we also have
$$
\begin{equation}
\omega(\Theta)\geqslant \frac{m\omega(\Theta^\top)+m-1}{(n-1)\omega(\Theta^\top)+n}\,.
\end{equation}
\tag{77}
$$
As we mentioned in § 3.3.1, in the case when either $m=1$ or $n=1$, the sharpness of (75) was proved by Jarník in [32] and [33]. For $\min(n,m)>1$ the sharpness of (75) has been established only if
$$
\begin{equation*}
\omega(\Theta)\geqslant\max\biggl(\frac{m}{n}\,,\frac{n-1}{m-1}\biggr)
\end{equation*}
\notag
$$
(in particular, if $m\geqslant n$). This result is also due to Jarník [72]. In all the remaining cases the sharpness of (75) remains unproved. 4.3.2. Inequalities for uniform exponentsJarník’s transference inequalities, that is, (19) and (21), were generalised in 1951 by Apfelbeck [73] to the case of arbitrary $n$ and $m$. He proved the ‘uniform’ analogue of (75),
$$
\begin{equation}
\widehat\omega(\Theta^\top)\geqslant \frac{n\widehat\omega(\Theta)+n-1}{(m-1)\widehat\omega(\Theta)+m}\,,
\end{equation}
\tag{78}
$$
and also the inequality
$$
\begin{equation}
\widehat\omega(\Theta^\top)\geqslant\frac{1}{m} \biggl(n+\frac{n(n\widehat\omega(\Theta)-m)-2n(m+n-3)} {(m-1)(n\widehat\omega(\Theta)-m)+m-(m-2)(m+n-3)}\biggr)
\end{equation}
\tag{79}
$$
under the assumption that $m>1$ and $\widehat\omega(\Theta)>(2(m+n-1)(m+n-3)+m)/n$. In 2012 inequalities (78) and (79) were improved by this author in [36] and [37]. The following theorem generalises Theorem 13 to the case of arbitrary $n$ and $m$. Note that, in general, $\widehat\omega(\Theta)$ and $\widehat\omega(\Theta^\top)$ can be equal to $+\infty$, and this gives meaning to (80) when some denominators vanish. If $\min(n,m)>1$, then the question of whether (80) is sharp remains open. We recall that in the case when either $m=1$ or $n=1$ the sharpness of (80) was proved by Marnat [38] and (independently) Schmidt and Summerer [39]. 4.3.3. ‘Mixed’ inequalitiesIn the same papers [36] and [37] the following theorem was proved. It generalises Theorem 14 of Laurent and Bugeaud and refines Theorem 28 of Dyson. Theorem 30 (German, 2012). Given $\Theta\in\mathbb{R}^{n\times m}$, $m+n\geqslant3$, assume that the space of rational solutions of the equation is not one-dimensional. Then Clearly, (81) coincides with (75). This inequality is stronger than (82) and (83) if and only if
$$
\begin{equation*}
\widehat\omega(\Theta)<\min\biggl(\frac{(m-1)\omega(\Theta)+m}{m+n-1}\,, \frac{(m+n-1)\omega(\Theta)}{n-1+n\omega(\Theta)}\biggr).
\end{equation*}
\notag
$$
For instance, if $\widehat\omega(\Theta)<(m+n-1)/n$ and $\omega(\Theta)$ is large enough. Thus, if $\min(n,m)>1$, inequalities (82) and (83) are not guaranteed to improve upon Dyson’s inequality. Their sharpness is also doubtful, for, as we noted in § 3.3.3, they are not sharp for $m=1$, $n\geqslant3$ and for $n=1$, $m\geqslant3$. We note also that, so far, no analogue of Schmidt–Summerer’s inequality (25) is known in the case when $\min(n,m)>1$. It would be very interesting to find such an analogue, which, in combination with (80), would give (82) and (83) — just as (25) in combination with (22) gives (24). 4.3.4. Inequalities between regular and uniform exponentsIn [48] Jarník obtained the following result. Theorem 31 ( Jarník, 1954). Given $\Theta\in\mathbb{R}^{n\times m}$, assume that the equation admits no non-zero integer solutions. Then (i) for $m=2$ we have
$$
\begin{equation}
\frac{\omega(\Theta)}{\widehat\omega(\Theta)}\geqslant \widehat\omega(\Theta)-1;
\end{equation}
\tag{84}
$$
(ii) for $m\geqslant3$ and $\widehat\omega\geqslant(5m^2)^{m-1}$ we have It is easy to see that (84) is stronger than the trivial estimate $\omega(\Theta)\geqslant\widehat\omega(\Theta)$ only for $\widehat\omega(\Theta)>2$. In 2013 Moshchevitin [74] improved Jarník’s result for $m=2$ and $n\geqslant2$. His inequality is stronger than (84) and the trivial inequality for each $\widehat\omega(\Theta)>1$. Theorem 32 (Moshchevitin, 2013). Let $\Theta\in\mathbb{R}^{n\times 2}$, $n\geqslant2$. Assume that among the rows of the matrix $\Theta$ there are two rows that are linearly independent with the vectors $(1,0)$ and $(0,1)$ over $\mathbb{Q}$. Also assume that $\widehat\omega(\Theta)\geqslant1$. Then For $m=2$ and $n=2$ Theorem 32 is not the first improvement of (84). In his survey [31] Moshchevitin presented an improvement of (84), however, it was an improvement indeed only for $1<\widehat\omega(\Theta)<(3+\sqrt5\,)/2$. Theorem 32 is stronger than that result. It is worth mentioning that Schmidt noticed that the statement of Theorem 32 in Moshchevitin’s paper [74] contains two misprints. First, Moshchevitin’s proof works for every $n\geqslant2$, while in the statement of the theorem the restriction $n\geqslant3$ is imposed. Second, there is a misprint in the formula defining the constant $G(\Theta)$. There are currently no estimates for the ratio $\omega(\Theta)/\widehat\omega(\Theta)$ in the case when $\min(n,m)>1$ which are stronger than Theorems 31 and 32 of Jarník and Moshchevitin. 4.4. Embedding into $ {\mathbb{R}}^{m+n}$Let us demonstrate how to use the constructions discussed in § 3.4 to prove transference theorems. We generalise the approach described in § 3.4.3 by ‘embedding’ the problems corresponding to $\Theta$ and $\Theta^\top$ into an $(m+n)$-dimensional Euclidean space. Set Consider the lattices
$$
\begin{equation}
\Lambda=\Lambda(\Theta)=\begin{pmatrix} \mathbf I_m & \\ -\Theta & \mathbf I_n \end{pmatrix}\mathbb{Z}^d\quad\text{and}\quad \Lambda^\ast=\Lambda^\ast(\Theta)= \begin{pmatrix} \mathbf I_m & \Theta^\top \\ & \mathbf I_n \end{pmatrix}\mathbb{Z}^d
\end{equation}
\tag{85}
$$
and the two families of parallelepipeds
$$
\begin{equation}
\mathcal{P}(t,\gamma) =\biggl\{\mathbf z=(z_1,\dots,z_d)\in \mathbb{R}^d \biggm| \begin{alignedat}{2} &|z_j|\leqslant t,&\quad j&=1,\dots,m \\ &|z_{m+i}|\leqslant t^{-\gamma},&\quad i&=1,\dots,n \end{alignedat} \biggr\}
\end{equation}
\tag{86}
$$
and
$$
\begin{equation}
\mathcal{Q}(s,\delta) =\biggl\{\mathbf z=(z_1,\dots,z_d)\in \mathbb{R}^d \biggm| \begin{alignedat}{2} &|z_j|\leqslant s^{-\delta},&\quad j&=1,\dots,m \\ &|z_{m+i}|\leqslant s,&\quad i&=1,\dots,n \end{alignedat} \biggr\}.
\end{equation}
\tag{87}
$$
Then
$$
\begin{equation}
\begin{aligned} \, \omega(\Theta) & =\sup\biggl\{ \gamma\geqslant\frac mn \biggm| \forall\,t_0\in\mathbb{R}\,\ \exists\,t>t_0\colon \ \ \mathcal{P}(t,\gamma) \cap\Lambda\ne \{\mathbf 0\} \biggr\}, \\ \widehat\omega(\Theta) & =\sup\biggl\{ \gamma\geqslant\frac mn \biggm| \exists\,t_0\in\mathbb{R}:\ \forall\,t>t_0\ \ \mathcal{P}(t,\gamma)\cap \Lambda\ne \{\mathbf 0\} \biggr\}, \\ \omega(\Theta^\top) & =\sup\biggl\{ \delta\geqslant\frac{n}{m}\biggm| \forall\,s_0\in\mathbb{R}\,\ \exists\,s>s_0\colon\ \ \mathcal{Q}(s,\delta) \cap\Lambda^\ast\ne \{\mathbf 0\} \biggr\}, \\ \widehat\omega(\Theta^\top) & =\sup\biggl\{ \delta\geqslant \frac{n}{m}\biggm| \exists\,s_0\in\mathbb{R}\colon \ \forall\,s>s_0\ \ \mathcal{Q}(s,\delta)\cap\Lambda^\ast\ne \{\mathbf 0\}\biggr\}. \end{aligned}
\end{equation}
\tag{88}
$$
4.4.1. Dyson’s theoremIf
$$
\begin{equation}
t=s^{((n-1)\delta+n)/(d-1)}\quad\text{and}\quad \gamma=\frac{m\delta+m-1}{(n-1)\delta+n}
\end{equation}
\tag{89}
$$
then $\mathcal{Q}(s,\delta)$ is the $(d-1)$th compound (see Definition 14 in § 3.4.2) of $\mathcal{P}(t,\gamma)$, that is, $\mathcal{Q}(s,\delta)=\mathcal{P}(t,\gamma)^\ast$. By Mahler’s theorem (once again, in disguise of Theorem 22) we have
$$
\begin{equation}
\mathcal{Q}(s,\delta)\cap\Lambda^\ast\ne \{\mathbf 0\}\ \ \implies\ \ (d-1)\mathcal{P}(t,\gamma)\cap\Lambda\ne \{\mathbf 0\}.
\end{equation}
\tag{90}
$$
Hence by (88)
$$
\begin{equation*}
\omega(\Theta^\top)\geqslant\delta\ \ \implies\ \ \omega(\Theta)\geqslant\gamma=\frac{m\delta+m-1}{(n-1)\delta+n}\,.
\end{equation*}
\notag
$$
We obtain
$$
\begin{equation*}
\omega(\Theta)\geqslant \frac{\omega(m\Theta^\top)+m-1}{(n-1)\omega(\Theta^\top)+n}\,.
\end{equation*}
\notag
$$
This is how inequality (77), and therefore Theorem 28 due to Dyson, is proved. 4.4.2. Badly approximable matricesThe correspondence (89) and the implication (90) provide a very simple proof of the fact that $\Theta$ is badly approximable if and only if so is $\Theta^\top$ (see Definition 21). Indeed, $\Theta$ is badly approximable if and only if there exists a positive constant $c$ such that for every $t>1$ the parallelepiped $c\mathcal{P}(t,m/n)$ contains no non-zero points of $\Lambda$. In a similar way, $\Theta^\top$ is badly approximable if and only if there exists a positive constant $c$ such that for every $s>1$ the parallelepiped $c\mathcal{Q}(s,n/m)$ contains no non-zero points of $\Lambda^\ast$. An application of (90) completes the argument. 4.4.3. The theorem on uniform exponentsTheorem 29 on uniform exponents can be proved with the help of the construction of ‘nodes’ and ‘leaves’ described in § 3.4.5. Now, instead of the parallelepipeds $\mathcal{Q}_r$ defined by (55), we consider the parallelepipeds
$$
\begin{equation*}
\mathcal{Q}_r=\biggl\{\,\mathbf z=(z_1,\dots,z_d) \in\mathbb{R}^d \biggm| \begin{aligned} \, &|z_j|\leqslant (hH/r)^{-\alpha},\quad\, j=1,\dots,m \\ &|z_{m+i}|\leqslant r,\qquad\qquad i=1,\dots,n \end{aligned} \biggr\}.
\end{equation*}
\notag
$$
Correspondingly, the parallelepipeds $\mathcal{P}(t,\gamma)$ and $\mathcal{Q}(s,\delta)$ are to be defined by (86) and (87), rather than by (47) and (48). Then we can use Fig. 8 unaltered, with the agreement that now $u$ and $v$ denote $\max(|z_{m+1}|,\dots,|z_d|)$ and $\max(|z_1|,\dots,|z_m|)$, respectively. Upon fixing $s>1$ and $\delta\geqslant n/m$, we set, as in (89),
$$
\begin{equation*}
t=s^{((n-1)\delta+n)/(d-1)}\quad\text{and}\quad \gamma=\frac{m\delta+m-1}{(n-1)\delta+n}\,.
\end{equation*}
\notag
$$
Also set
$$
\begin{equation*}
h=s,\qquad \beta=\delta,\quad\text{and}\quad \alpha=\begin{cases} \dfrac{(d-1)\delta}{m\delta+m-1} & \text{if}\ \delta\leqslant\dfrac{m-1}{n-1}\,, \\ \dfrac{(n-1)\delta+n\vphantom{1^{\big|}}}{d-1} & \text{if}\ \delta\geqslant\dfrac{m-1}{n-1}\,. \end{cases}
\end{equation*}
\notag
$$
For such a choice of the parameters the quantities $\gamma$ and $\alpha$ are related by
$$
\begin{equation}
\gamma=\begin{cases} \dfrac{m-1}{n-\alpha} & \text{if}\ \alpha\leqslant1, \\ \dfrac{m-\alpha^{-1}}{n-1} & \text{if}\ \alpha\geqslant1. \end{cases}
\end{equation}
\tag{91}
$$
Arguing in the manner of § 3.4.6, we obtain the implication
$$
\begin{equation*}
\widehat\omega(\Theta^\top)\geqslant\alpha\ \ \implies\ \ \widehat\omega(\Theta)\geqslant\gamma,
\end{equation*}
\notag
$$
which shows, in view of (91), that
$$
\begin{equation*}
\widehat\omega(\Theta)\geqslant \begin{cases} \dfrac{m-1}{n-\widehat\omega(\Theta^\top)} & \text{if}\ \widehat\omega(\Theta^\top)\leqslant1, \\ \dfrac{m-\widehat\omega(\Theta^\top)^{-1}}{n-1} & \text{if}\ \widehat\omega(\Theta^\top)\geqslant1. \end{cases}
\end{equation*}
\notag
$$
Swapping the triple $(n,m,\Theta)$ for the triple $(m,n,\Theta^\top)$, we arrive at (80). This is how Theorem 29 is proved. 4.4.4. Parametric geometry of numbersLet us interpret the problem of approximating zero with values of several linear forms in the spirit of parametric geometry of numbers. In accordance with § 3.4.8, let us choose a lattice and a path. Define lattices $\Lambda$ and $\Lambda^\ast$ by (85). Define paths $\mathfrak{T}$ and $\mathfrak{T}^\ast$ by the maps
$$
\begin{equation*}
\begin{gathered} \, s\mapsto\boldsymbol{\tau}(s)=(\tau_1(s),\dots,\tau_d(s)), \\ \tau_1(s)=\cdots=\tau_m(s)=s,\quad \tau_{m+1}(s)=\cdots=\tau_d(s)=-\frac{ms}{n}\,, \end{gathered}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\begin{gathered} \, s\mapsto\boldsymbol{\tau}^\ast(s)=\bigl(\tau^\ast_1(s),\dots,\tau^\ast_d(s)\bigr), \\ \tau^\ast_1(s)=\cdots=\tau^\ast_m(s)=-\frac{ns}{m}\,,\quad \tau^\ast_{m+1}(s)=\cdots=\tau^\ast_d(s)=s, \end{gathered}
\end{equation*}
\notag
$$
respectively. Then the regular and uniform Diophantine exponents can be expressed in terms of Schmidt–Summerer exponents with the help of the relations
$$
\begin{equation}
\begin{gathered} \, \bigl(1+\omega(\Theta)\bigr) \bigl(1+\underline{\varphi}_1(\Lambda,\mathfrak{T})\bigr)= \bigl(1+\widehat\omega(\Theta)\bigr) \bigl(1+\overline{\varphi}_1(\Lambda,\mathfrak{T})\bigr)=\frac{d}{n}\,, \\ \bigl(1+\omega(\Theta^\top)\bigr) \bigl(1+\underline{\varphi}_1(\Lambda^\ast,\mathfrak{T}^\ast)\bigr)= \bigl(1+\widehat\omega(\Theta^\top)\bigr) \bigl(1+\overline{\varphi}_1(\Lambda^\ast,\mathfrak{T}^\ast)\bigr)=\frac{d}{m}\,. \end{gathered}
\end{equation}
\tag{92}
$$
They are deduced, just as relations (67), directly from the definitions (see [37]). Since the functions $L_k(\boldsymbol{\tau})$ and $S_k(\boldsymbol{\tau})$ enjoy properties (i) – (iv) formulated in § 3.4.8 for any fixed lattice, we also have
$$
\begin{equation*}
S_1(\Lambda,\boldsymbol{\tau})\leqslant \frac{S_{d-1}(\Lambda,\boldsymbol{\tau})}{d-1}= \frac{S_1(\Lambda^\ast,-\boldsymbol{\tau})}{d-1}+O(1).
\end{equation*}
\notag
$$
For Schmidt–Summerer exponents this gives
$$
\begin{equation}
\underline{\Phi}_1(\Lambda,\mathfrak{T})\leqslant \frac{\underline{\Phi}_{d-1}(\Lambda,\mathfrak{T})}{d-1}= \frac{n}{m(d-1)}\,\underline{\Phi}_1(\Lambda^\ast,\mathfrak{T}^\ast),
\end{equation}
\tag{93}
$$
since $\boldsymbol{\tau}^\ast(s)=-(n/m)\boldsymbol{\tau}(s)$. Taking the inequalities
$$
\begin{equation*}
\underline{\Phi}_1(\Lambda,\mathfrak{T})= \underline{\varphi}_1(\Lambda,\mathfrak{T})\quad\text{and}\quad \underline{\Phi}_1(\Lambda^\ast,\mathfrak{T}^\ast)= \underline{\varphi}_1(\Lambda^\ast,\mathfrak{T}^\ast),
\end{equation*}
\notag
$$
into account we obtain
$$
\begin{equation}
\underline{\varphi}_1(\Lambda,\mathfrak{T})\leqslant \dfrac{n}{m(d-1)}\,\underline{\varphi}_1(\Lambda^\ast,\mathfrak{T}^\ast).
\end{equation}
\tag{94}
$$
Rewriting (94), with the help of (92), in terms of $\omega(\Theta)$ and $\omega(\Theta^\top)$ leads exactly to Dyson’s inequality (75). Thus, parametric geometry of numbers provides another way to prove Dyson’s theorem. And just as with Khintchine’s inequalities, this method enables one to refine Dyson’s inequality by splitting it into a chain of inequalities between intermediate exponents. The explicit formulation of this result, as well as its proof, can be found in [37].
5. Multiplicative exponentsAs in the previous section, given a matrix
$$
\begin{equation*}
\Theta=\begin{pmatrix} \theta_{11} & \dots & \theta_{1m} \\ \vdots & \ddots & \vdots \\ \theta_{n1} & \dots & \theta_{nm} \end{pmatrix},\qquad \theta_{ij}\in\mathbb{R},\quad m+n\geqslant3,
\end{equation*}
\notag
$$
consider the same system of linear equations with variables $\mathbf{x}=(x_1,\dots,x_m)\in\mathbb{R}^m$ and $\mathbf{y}=(y_1,\dots,y_n)\in\mathbb{R}^n$. Until now, we have been dealing with the question of how small the quantity $|\Theta\mathbf{x}-\mathbf{y}|$ can be for $\mathbf{x}\in\mathbb{Z}^m$ and $\mathbf{y}\in\mathbb{Z}^n$ satisfying the restriction $0<|\mathbf{x}|\leqslant t$, where $|\,{\cdot}\,|$ denotes the $\ell^\infty$-norm (sup-norm). The choice of the $\ell^\infty$-norm for estimating the ‘magnitude’ of a vector is quite conditional. Considering $\ell^2$-norm alters the problem slightly preserving, however, the values of Diophantine exponents. The reason is that any two norms in a finite dimensional space are known to be equivalent. The problem changes essentially if a vector’s ‘magnitude’ is measured in terms of the product of its coordinates. If we associate the $\ell^1$-norm with the arithmetic mean, then the product of coordinates can be associated with the geometric mean. Such an approach is also absolutely classical. For example, we can recall the famous Littlewood conjecture. Littlewood’s Conjecture. For any $\theta_1,\theta_2\in\mathbb{R}$ and any $\varepsilon>0$ the inequality admits infinitely many solutions in $(x,y_1,y_2)\in\mathbb{Z}^3$ with non-zero $x$. Correspondingly, it is natural to consider, as a multiplicative analogue of the regular Diophantine exponent of a pair $(\theta_1,\theta_2)$, the supremum of real $\gamma$ such that the inequality
$$
\begin{equation*}
\prod_{i=1,2}|\theta_ix-y_i|^{1/2} \leqslant |x|^{-\gamma}
\end{equation*}
\notag
$$
admits infinitely many solutions in $(x,y_1,y_2)\in\mathbb{Z}^3$ with non-zero $x$. This is an example for the case $n=2$, $m=1$. In order to give the definition of multiplicative exponents in the general case, set for each $\mathbf{z}=(z_1,\dots,z_k)\in\mathbb{R}^k$ set
$$
\begin{equation*}
\Pi(\mathbf z)=\prod_{1\leqslant i\leqslant k}|z_i|^{1/k}\quad\text{and}\quad \Pi'(\mathbf z)=\prod_{1\leqslant i\leqslant k}(\max(1,|z_i|))^{1/k}.
\end{equation*}
\notag
$$
Definition 22. The supremum of real $\gamma$ satisfying the condition that there exist arbitrarily large $t$ such that (respectively, for every $t$ large enough) the system admits a non-zero solution $(\mathbf{x},\mathbf{y})\in\mathbb{Z}^m\oplus\mathbb{Z}^n$ with non-zero $\mathbf{x}$ is called the regular (respectively, uniform) multiplicative Diophantine exponent of $\Theta$ and is denoted by $\omega_\times(\Theta)$ (respectively, $\widehat\omega_\times(\Theta)$). Ordinary and multiplicative exponents are related by the inequalities
$$
\begin{equation}
\begin{aligned} \, \omega(\Theta)& \leqslant \omega_\times(\Theta)\leqslant \begin{cases} m\omega(\Theta) & \text{if}\ n=1, \\ +\infty & \text{if}\ n\geqslant2, \end{cases} \\ \widehat\omega(\Theta) &\leqslant \widehat\omega_\times(\Theta)\leqslant \begin{cases} m\widehat\omega(\Theta) & \text{if}\ n=1, \\ +\infty & \text{if}\ n\geqslant2, \end{cases} \end{aligned}
\end{equation}
\tag{96}
$$
which follow from the fact that for each $\mathbf{z}\in\mathbb{R}^k$ we have and for each $\mathbf{z}\in\mathbb{Z}^k$ we have
$$
\begin{equation*}
|\mathbf z|^{1/k}\leqslant\Pi'(\mathbf z)\leqslant|\mathbf z|.
\end{equation*}
\notag
$$
Corollary 11 to Dirichlet’s theorem implies the ‘trivial’ inequalities
$$
\begin{equation*}
\begin{alignedat}{2} \omega_\times(\Theta)&\geqslant\omega(\Theta)\geqslant \frac{m}{n}\,,&\qquad \omega_\times(\Theta^\top)&\geqslant\omega(\Theta^\top)\geqslant \frac{n}{m}\,, \\ \widehat\omega_\times(\Theta)&\geqslant\widehat\omega(\Theta)\geqslant \frac{m}{n}\,,&\qquad \widehat\omega_\times(\Theta^\top)&\geqslant\widehat\omega(\Theta^\top) \geqslant \frac{n}{m}\,, \end{alignedat}
\end{equation*}
\notag
$$
where $\Theta^\top$, as before, denotes the transposed matrix. 5.1. An analogue of Dyson’s theoremIn 1979 Schmidt and Wang [75] showed that, just as in the case of ordinary Diophantine exponents (see (76)), we have
$$
\begin{equation}
\omega_\times(\Theta)=\frac{m}{n}\ \ \iff \ \ \omega_\times(\Theta^\top)=\frac{n}{m}\,.
\end{equation}
\tag{97}
$$
Again, this is a manifestation of the transference principle. As Bugeaud [76] noted, the argument Schmidt and Wang used to prove (97) can be applied to prove an analogue of Dyson’s inequality (75) under certain restrictions imposed on $\Theta$. An analogue with no restriction on $\Theta$ was obtained by this author in [77]. The method by which (98) is proved gives the analogous inequality for uniform exponents:
$$
\begin{equation}
\widehat\omega_\times(\Theta^\top)\geqslant \frac{n\widehat\omega_\times(\Theta)+n-1} {(m-1)\widehat\omega_\times(\Theta)+m}\,.
\end{equation}
\tag{99}
$$
However, nothing stronger concerning uniform multiplicative exponents is currently known. The method used to prove Theorem 29 concerning ordinary uniform exponents does not work in the multiplicative setting because of the non-convexity of the functionals $\Pi(\,\cdot\,)$ and $\Pi'(\,\cdot\,)$. In addition, there exist neither ‘mixed’ inequalities in the spirit of Theorems 14, 15, and 30, nor inequalities relating regular and uniform exponents, not even in the case $m+n=3$. 5.2. Applying Mahler’s theorem and dimension reductionThe standard application of Mahler’s theorem — either in the form of Theorem 21, or in the form of Theorem 22 — is not enough to prove Theorem 33. The reason is that $\Pi'(\,\cdot\,)$ differs from $\Pi(\,\cdot\,)$. In the paper [75] mentioned above Schmidt and Wang evaded this obstacle using induction with respect to dimension. The same technique was used in [77]. Here we describe a slightly more explicit construction, which also requires dimension reduction. 5.2.1. Embedding into $\mathbb{R}^{m+n}$Working with the functionals $\Pi(\,\cdot\,)$ and $\Pi'(\,\cdot\,)$ requires considering more extended families of parallelepipeds than (86) and (87). As in § 4.4, let us set and consider the lattices
$$
\begin{equation}
\Lambda=\Lambda(\Theta)=\begin{pmatrix} \mathbf I_m & \\ -\Theta & \mathbf I_n \end{pmatrix}\mathbb{Z}^d\quad\text{and}\quad \Lambda^\ast=\Lambda^\ast(\Theta)=\begin{pmatrix} \mathbf I_m & \Theta^\top \\ & \mathbf I_n \end{pmatrix}\mathbb{Z}^d.
\end{equation}
\tag{100}
$$
For each tuple $(\boldsymbol{\lambda},\boldsymbol{\mu})=(\lambda_1,\dots,\lambda_m, \mu_1,\dots,\mu_n)\in\mathbb{R}_+^d$ we define the parallelepiped $\mathcal{P}(\boldsymbol{\lambda},\boldsymbol{\mu})$ by
$$
\begin{equation}
\mathcal{P}(\boldsymbol{\lambda},\boldsymbol{\mu})= \biggl\{\,\mathbf z=(z_1,\dots,z_d)\in\mathbb{R}^d \biggm| \begin{aligned} \, &|z_j|\leqslant\lambda_j,\quad\,\ j=1,\dots,m \\ &|z_{m+i}|\leqslant\mu_i,\ \ i=1,\dots,n \end{aligned} \biggr\}.
\end{equation}
\tag{101}
$$
Let us also define, for all positive $t$, $\gamma$, $s$, and $\delta$, the families
$$
\begin{equation}
\mathcal{F}(t,\gamma)=\Bigl\{\mathcal{P}(\boldsymbol{\lambda},\boldsymbol{\mu}) \bigm| \Pi(\boldsymbol{\lambda})=t,\ \Pi(\boldsymbol{\mu})=t^{-\gamma},\ \min_{1\leqslant j\leqslant m}\lambda_j\geqslant1\Bigr\}
\end{equation}
\tag{102}
$$
and
$$
\begin{equation}
\mathcal{G}(s,\delta)=\Bigl\{\mathcal{P}(\boldsymbol{\lambda},\boldsymbol{\mu}) \bigm| \Pi(\boldsymbol{\lambda})=s^{-\delta},\ \Pi(\boldsymbol{\mu})=s,\ \min_{1\leqslant i\leqslant n}\mu_i\geqslant1\Bigr\}.
\end{equation}
\tag{103}
$$
Every parallelepiped $\mathcal{P}(\boldsymbol{\lambda},\boldsymbol{\mu})$ satisfying the conditions
$$
\begin{equation}
\Pi'(\boldsymbol{\lambda})\leqslant t \quad\text{and}\quad \Pi(\boldsymbol{\mu})\leqslant t^{-\gamma},
\end{equation}
\tag{104}
$$
is contained in a parallelepiped from (102). Conversely, each parallelepiped $\mathcal{P}(\boldsymbol{\lambda},\boldsymbol{\mu})$ from (102) satisfies (104). In a similar way, every parallelepiped $\mathcal{P}(\boldsymbol{\lambda},\boldsymbol{\mu})$ satisfying the conditions
$$
\begin{equation}
\Pi(\boldsymbol{\lambda})\leqslant s^{-\delta}, \qquad \Pi'(\boldsymbol{\mu})\leqslant s,
\end{equation}
\tag{105}
$$
is contained in a parallelepiped from (103). Conversely, each parallelepiped $\mathcal{P}(\boldsymbol{\lambda},\boldsymbol{\mu})$ from (103) satisfies (105). Therefore, the following analogue of (49) and (88) holds for multiplicative exponents:
$$
\begin{equation}
\begin{aligned} \, \omega_\times(\Theta)&=\sup\biggl\{\gamma\geqslant\frac{m}{n} \biggm| \forall\,t_0\in\mathbb{R}\,\ \exists\,t>t_0\colon \exists\mathcal{P}\in\mathcal{F}(t,\gamma)\colon \mathcal{P}\cap \Lambda\ne \{\mathbf 0\} \biggr\}, \\ \omega_\times(\Theta^\top)&=\sup\biggl\{\delta\geqslant\frac{n}{m}\biggm| \forall\,s_0\in\mathbb{R}\,\ \exists\,s>s_0\colon \exists\mathcal{P}\in\mathcal{G}(s,\delta)\colon \mathcal{P}\cap\Lambda^\ast\ne \{\mathbf 0\}\biggr\}. \end{aligned}
\end{equation}
\tag{106}
$$
5.2.2. PseudocompoundsTo each tuple
$$
\begin{equation*}
(\boldsymbol{\lambda},\boldsymbol{\mu})= (\lambda_1,\dots,\lambda_m,\mu_1,\dots,\mu_n)\in\mathbb{R}_+^d
\end{equation*}
\notag
$$
we assign the tuple
$$
\begin{equation*}
(\boldsymbol{\lambda}^\ast,\boldsymbol{\mu}^\ast)= (\lambda_1^\ast,\dots,\lambda_m^\ast,\mu_1^\ast,\dots,\mu_n^\ast),
\end{equation*}
\notag
$$
defined as follows:
$$
\begin{equation}
\begin{alignedat}{2} \lambda_j^\ast &=\lambda_j^{-1}\Pi(\boldsymbol{\lambda})^m\Pi(\boldsymbol{\mu})^n,&\qquad j&=1,\dots,m, \\ \mu_i^\ast&=\mu_i^{-1}\Pi(\boldsymbol{\lambda})^m\Pi(\boldsymbol{\mu})^n,&\qquad i&=1,\dots,n. \end{alignedat}
\end{equation}
\tag{107}
$$
Then $\mathcal{P}(\boldsymbol{\lambda},\boldsymbol{\mu})^\ast =\mathcal{P}(\boldsymbol{\lambda}^\ast,\boldsymbol{\mu}^\ast)$, that is, $\mathcal{P}(\boldsymbol{\lambda}^\ast,\boldsymbol{\mu}^\ast)$ is the pseudocompound of $\mathcal{P}(\boldsymbol{\lambda},\boldsymbol{\mu})$ (see Definition 14 in § 3.4.2). As in § 4.4.1, let us assign to each pair $(s,\delta)$ the pair $(t,\gamma)$ defined by (89), that is,
$$
\begin{equation}
t=s^{((n-1)\delta+n)/(d-1)}\quad\text{and}\quad \gamma=\frac{m\delta+m-1}{(n-1)\delta+n}.
\end{equation}
\tag{108}
$$
Consider the family
$$
\begin{equation*}
\mathcal{F}^\ast(t,\gamma)=\{\mathcal{P}^\ast \mid \mathcal{P}\in\mathcal{F}(t,\gamma)\}
\end{equation*}
\notag
$$
of pseudocompounds. If $\mathcal{P}(\boldsymbol{\lambda},\boldsymbol{\mu}) \in\mathcal{F}(t,\gamma)$, then
$$
\begin{equation*}
\begin{aligned} \, \Pi(\boldsymbol{\lambda}^\ast)&=\Pi(\boldsymbol{\lambda})^{m-1}\Pi(\boldsymbol{\mu})^n= t^{(m-1)-n\gamma}=s^{-\delta}, \\ \Pi(\boldsymbol{\mu}^\ast)&=\Pi(\boldsymbol{\lambda})^m\Pi(\boldsymbol{\mu})^{n-1}= t^{m-(n-1)\gamma}=s \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\max_{1\leqslant j\leqslant m}\lambda_j^\ast\leqslant \Pi(\boldsymbol{\lambda})^m\Pi(\boldsymbol{\mu})^n.
\end{equation*}
\notag
$$
For convenience, let us set
$$
\begin{equation}
\pi(\boldsymbol{\lambda},\boldsymbol{\mu})=\Pi(\boldsymbol{\lambda})^m\Pi(\boldsymbol{\mu})^n.
\end{equation}
\tag{109}
$$
Then $\mathcal{F}^\ast(t,\gamma)$ can be written as
$$
\begin{equation}
\mathcal{F}^\ast(t,\gamma)=\Bigl\{\mathcal{P}(\boldsymbol{\lambda},\boldsymbol{\mu}) \bigm| \Pi(\boldsymbol{\lambda})=s^{-\delta},\ \Pi(\boldsymbol{\mu})=s,\ \max_{1\leqslant j\leqslant m} \lambda_j\leqslant\pi(\boldsymbol{\lambda},\boldsymbol{\mu})\Bigr\}.
\end{equation}
\tag{110}
$$
As we can see, the inclusion $\mathcal{F}^\ast(t,\gamma) \subset\mathcal{G}(s,\delta)$ does not hold because of the absence of the condition $\min_{1\leqslant i\leqslant n}\mu_i\geqslant1$ in (110). Therefore, we consider more narrow families
$$
\begin{equation}
\begin{aligned} \, \mathcal{G}'(s,\delta)&=\Bigl\{\mathcal{P}(\boldsymbol{\lambda},\boldsymbol{\mu})\in \mathcal{F}^\ast(t,\gamma)\bigm| \min_{1\leqslant i\leqslant n}\mu_i\geqslant1\Bigr\} \nonumber \\ &=\Bigl\{\mathcal{P}(\boldsymbol{\lambda},\boldsymbol{\mu})\in\mathcal{G}(s,\delta) \bigm| \max_{1\leqslant j\leqslant m}\lambda_j\leqslant \pi(\boldsymbol{\lambda},\boldsymbol{\mu})\Bigr\} \nonumber \\ &=\left\{\vphantom{\Biggm\}}\right. \mathcal{P}(\boldsymbol{\lambda},\boldsymbol{\mu}) \biggm| \begin{aligned} \, &\Pi(\boldsymbol{\lambda})=s^{-\delta},\ \ \displaystyle\max_{1\leqslant j\leqslant m}\lambda_j\leqslant \pi(\boldsymbol{\lambda},\boldsymbol{\mu})\vphantom{1^{\big|}} \\ &\Pi(\boldsymbol{\mu})=s,\quad\ \ \displaystyle \min_{1\leqslant i\leqslant n}\mu_i\geqslant1 \end{aligned} \left.\vphantom{\Biggm|} \right\} \end{aligned}
\end{equation}
\tag{111}
$$
and
$$
\begin{equation}
\begin{aligned} \, \mathcal{F}'(t,\gamma) & =\biggl\{\mathcal{P}(\boldsymbol{\lambda},\boldsymbol{\mu})\in \mathcal{F}(t,\gamma) \bigm| \max_{1\leqslant i\leqslant n}\mu_i\leqslant \pi(\boldsymbol{\lambda},\boldsymbol{\mu})\biggr\} \nonumber \\ &=\left\{\vphantom{\Biggm\}}\right. \mathcal{P}(\boldsymbol{\lambda},\boldsymbol{\mu})\biggm| \begin{aligned} \, &\Pi(\boldsymbol{\lambda})=t,\quad\,\ \displaystyle\min_{1\leqslant j\leqslant m}\lambda_j\geqslant1 \vphantom{1^{\big|}} \\ &\Pi(\boldsymbol{\mu})=t^{-\gamma},\ \ \displaystyle\max_{1\leqslant i\leqslant n}\mu_i\leqslant \pi(\boldsymbol{\lambda},\boldsymbol{\mu}) \end{aligned}\left.\vphantom{\Biggm|} \right\}.\ \ \end{aligned}
\end{equation}
\tag{112}
$$
Then
$$
\begin{equation*}
\mathcal{G}'(s,\delta)=\{\mathcal{P}^\ast \mid \mathcal{P}\in\mathcal{F}'(t,\gamma)\}.
\end{equation*}
\notag
$$
Thus, Mahler’s theorem in disguise of Theorem 22 gives us the implication
$$
\begin{equation}
\exists\mathcal{P}\in\mathcal{G}'(s,\delta)\colon\mathcal{P}\cap \Lambda^\ast\ne\{\mathbf 0\}\ \ \implies\ \ \exists\mathcal{P}\in \mathcal{F}'(t,\gamma)\colon(d-1)\mathcal{P}\cap\Lambda\ne\{\mathbf 0\}.
\end{equation}
\tag{113}
$$
But the proof of Theorem 33 requires the analogous implication for the families $\mathcal{G}(s,\delta)$ and $\mathcal{F}(t,\gamma)$. Note, however, that it suffices to show that we can replace $\mathcal{G}'(s,\delta)$ by $\mathcal{G}(s,\delta)$ in (113), since $\mathcal{F}'(t,\gamma)\subset\mathcal{F}(t,\gamma)$. 5.2.3. Dimension reductionLet us show that we can indeed improve (113) in the way indicated above, that is, that the following statement holds. Theorem 34. Let $\Lambda$ and $\Lambda^\ast$ be defined by (100). Let $t$, $\gamma$, $s$, and $\delta$ be positive real numbers related by (108). Let $\mathcal{F}(s,\delta)$, $\mathcal{G}(s,\delta)$, $\mathcal{F}'(t,\gamma)$, and $\mathcal{G}'(s,\delta)$ be defined by (102), (103), (111), and (112). Then It is clear that for $m=1$ the families $\mathcal{G}(s,\delta)$ and $\mathcal{G}'(s,\delta)$ coincide, since $s>1$. Assume that $m\geqslant2$. Consider arbitrary tuples
$$
\begin{equation*}
\boldsymbol{\lambda}=(\lambda_1,\dots,\lambda_m)\in\mathbb{R}_+^m\quad\text{and}\quad \boldsymbol{\mu}=(\mu_1,\dots,\mu_n)\in\mathbb{R}_+^n
\end{equation*}
\notag
$$
satisfying the conditions
$$
\begin{equation}
\Pi(\boldsymbol{\lambda})=t\quad\text{and}\quad \Pi(\boldsymbol{\mu})=t^{-\gamma},
\end{equation}
\tag{115}
$$
and let $\boldsymbol{\lambda}^\ast$ and $\boldsymbol{\mu}^\ast$ be defined by (107). Assume that
$$
\begin{equation}
\mathcal{P}(\boldsymbol{\lambda}^\ast,\boldsymbol{\mu}^\ast)\in \mathcal{G}(s,\delta)\setminus\mathcal{G}'(s,\delta).
\end{equation}
\tag{116}
$$
Changing the ordering, if required, we can assume that Then by (107)
$$
\begin{equation*}
\lambda_1^\ast\geqslant\cdots\geqslant\lambda_m^\ast.
\end{equation*}
\notag
$$
It follows from (116) that $\lambda_1^\ast>\pi(\boldsymbol{\lambda},\boldsymbol{\mu})$. Hence by (107) we also have $\lambda_1<1$. Let $k$ be the greatest index such that $\lambda_1\cdots\lambda_k<1$. Since $t>1$, it follows from the relation $\Pi(\boldsymbol{\lambda})=t$ that $k<m$. Taking (107) into account we obtain
$$
\begin{equation}
\begin{alignedat}{2} \lambda_1\cdots\lambda_k&<1,&\qquad \lambda_m &\geqslant\cdots\geqslant\lambda_{k+1}\geqslant1, \\ \lambda_1\cdots\lambda_k\lambda_{k+1}^{m-k}&\geqslant1,&\qquad \lambda^\ast_m &\leqslant\cdots\leqslant\lambda^\ast_{k+1}\leqslant \pi(\boldsymbol{\lambda},\boldsymbol{\mu}). \end{alignedat}
\end{equation}
\tag{117}
$$
Along with the tuple $\boldsymbol{\lambda}$ consider the tuple $\widehat{\boldsymbol{\lambda}}=(\widehat\lambda_1,\dots,\widehat\lambda_m)$, where
$$
\begin{equation}
\begin{alignedat}{2} \widehat\lambda_j&=1, &\qquad j&=1,\dots,k, \\ \widehat\lambda_j&=\lambda_j(\lambda_1\cdots\lambda_k)^{1/(m-k)},&\qquad j&=k+1,\dots,m. \end{alignedat}
\end{equation}
\tag{118}
$$
Then by (115)–(117) we have
$$
\begin{equation*}
\Pi(\widehat{\boldsymbol{\lambda}})=t,\quad \Pi(\boldsymbol{\mu})=t^{-\gamma},\quad \min_{1\leqslant j\leqslant m}\widehat\lambda_j\geqslant1,\quad\text{and}\quad \max_{1\leqslant i\leqslant n}\mu_i\leqslant \pi(\widehat{\boldsymbol{\lambda}},\boldsymbol{\mu}),
\end{equation*}
\notag
$$
that is,
$$
\begin{equation}
\mathcal{P}(\widehat{\boldsymbol{\lambda}},\boldsymbol{\mu})\in\mathcal{F}'(t,\gamma).
\end{equation}
\tag{119}
$$
Let us show that
$$
\begin{equation}
\mathcal{P}(\boldsymbol{\lambda}^\ast,\boldsymbol{\mu}^\ast)\cap\Lambda^\ast\ne \{\mathbf 0\} \ \ \implies\ \ (d-1)\mathcal{P}(\widehat{\boldsymbol{\lambda}},\boldsymbol{\mu})\cap\Lambda\ne \{\mathbf 0\}.
\end{equation}
\tag{120}
$$
Consider the truncated tuples
$$
\begin{equation*}
\boldsymbol{\lambda}_\downarrow=(\lambda_{k+1},\dots,\lambda_{m}), \qquad \boldsymbol{\lambda}^\ast_\downarrow= (\lambda^\ast_{k+1},\dots,\lambda^\ast_{m}),\quad\text{and} \quad \widehat{\boldsymbol{\lambda}}_\downarrow= (\widehat\lambda_{k+1},\dots,\widehat\lambda_m).
\end{equation*}
\notag
$$
Then
$$
\begin{equation*}
\mathcal{P}(\widehat{\boldsymbol{\lambda}}_\downarrow,\boldsymbol{\mu})^\ast= \biggl\{\,(z_{k+1},\dots,z_d)\in\mathbb{R}^{d-k} \biggm| \begin{alignedat}{2} &|z_j|\leqslant c\lambda_j^\ast,&\quad j&=k+1,\dots,m \\ &|z_{m+i}|\leqslant\mu_i^\ast,&\quad i&=1,\dots,n \end{alignedat} \biggr\},
\end{equation*}
\notag
$$
where $c=(\lambda_1\cdots\lambda_k)^{-1/(m-k)}$. Since $c>1$, we have
$$
\begin{equation}
\mathcal{P}(\boldsymbol{\lambda}_\downarrow^\ast,\boldsymbol{\mu}^\ast)\subset \mathcal{P}(\widehat{\boldsymbol{\lambda}}_\downarrow,\boldsymbol{\mu})^\ast.
\end{equation}
\tag{121}
$$
Consider also the matrix
$$
\begin{equation*}
\Theta_\downarrow=\begin{pmatrix} \theta_{1,k+1} & \dots & \theta_{1m} \\ \vdots & \ddots & \vdots \\ \theta_{n,k+1} & \dots & \theta_{nm} \end{pmatrix}
\end{equation*}
\notag
$$
obtained from $\Theta$ by deleting the first $k$ columns, and the lattices
$$
\begin{equation*}
\Lambda_\downarrow=\begin{pmatrix} \mathbf I_{m-k}& \\ -\Theta_\downarrow & \mathbf I_n \end{pmatrix}\mathbb{Z}^{d-k}\quad\text{and}\quad \Lambda^\ast_\downarrow=\begin{pmatrix} \mathbf I_{m-k} & \Theta^\top_\downarrow \\ & \mathbf I_n \end{pmatrix}\mathbb{Z}^{d-k}.
\end{equation*}
\notag
$$
It can easily be verified that the set
$$
\begin{equation*}
\{(0,\dots,0,z_{k+1},\dots,z_d)\in\mathbb{R}^d \mid (z_{k+1},\dots,z_d)\in\Lambda_\downarrow \}
\end{equation*}
\notag
$$
is a sublattice of $\Lambda$, and the set
$$
\begin{equation*}
\{(0,\dots,0,z_{k+1},\dots,z_d)\in\mathbb{R}^d \mid (z_{k+1},\dots,z_d)\in\Lambda^\ast_\downarrow\}
\end{equation*}
\notag
$$
is the projection of $\Lambda^\ast$ onto the plane of coordinates $z_{k+1},\dots,z_d$. Thus we obtain the implications
$$
\begin{equation}
\begin{aligned} \, \mathcal{P}(\boldsymbol{\lambda}^\ast,\boldsymbol{\mu}^\ast)\cap\Lambda^\ast\ne \{\mathbf 0\} &\ \ \implies\ \ \mathcal{P} (\boldsymbol{\lambda}^\ast_\downarrow,\boldsymbol{\mu}^\ast)\cap\Lambda^\ast_\downarrow \ne \{\mathbf 0\}, \\ \mathcal{P}(\widehat{\boldsymbol{\lambda}}_\downarrow,\boldsymbol{\mu})\cap \Lambda_\downarrow\ne \{\mathbf 0\}&\ \ \implies\ \ \mathcal{P}(\widehat{\boldsymbol{\lambda}},\boldsymbol{\mu})\cap\Lambda\ne \{\mathbf 0\}. \end{aligned}
\end{equation}
\tag{122}
$$
Finally, applying Mahler’s theorem in disguise of Theorem 22 we obtain the implication
$$
\begin{equation}
\mathcal{P}(\widehat{\boldsymbol{\lambda}}_\downarrow,\boldsymbol{\mu})^\ast\cap \Lambda^\ast_\downarrow\ne \{\mathbf 0\}\ \ \implies\ \ (d-k-1)\mathcal{P}(\widehat{\boldsymbol{\lambda}}_\downarrow,\boldsymbol{\mu})\cap \Lambda_\downarrow\ne \{\mathbf 0\}.
\end{equation}
\tag{123}
$$
Combining (121)–(123) we obtain the following chain of implications:
$$
\begin{equation*}
\begin{aligned} \, &\mathcal{P}(\boldsymbol{\lambda}^\ast,\boldsymbol{\mu}^\ast)\cap\Lambda^\ast\ne \{\mathbf 0\} \ \ \implies\ \ \mathcal{P}(\boldsymbol{\lambda}^\ast_\downarrow,\boldsymbol{\mu}^\ast)\cap \Lambda^\ast_\downarrow\ne \{\mathbf 0\} \\ &\qquad\implies\ \ \mathcal{P}(\widehat{\boldsymbol{\lambda}}_\downarrow,\boldsymbol{\mu})^\ast \cap\Lambda^\ast_\downarrow\ne \{\mathbf 0\}\ \ \implies\ \ (d-k-1)\mathcal{P}(\widehat{\boldsymbol{\lambda}}_\downarrow,\boldsymbol{\mu})\cap \Lambda_\downarrow\ne \{\mathbf 0\} \\ &\qquad\implies\ \ (d-k-1)\mathcal{P}(\widehat{\boldsymbol{\lambda}},\boldsymbol{\mu})\cap \Lambda\ne \{\mathbf 0\}\ \ \implies\ \ (d-1)\mathcal{P}(\widehat{\boldsymbol{\lambda}},\boldsymbol{\mu})\cap\Lambda\ne \{\mathbf 0\}. \end{aligned}
\end{equation*}
\notag
$$
Thus, (120) does hold indeed. Taking (119) into account we obtain
$$
\begin{equation}
\begin{aligned} \, &\exists\mathcal{P}\in\mathcal{G}(s,\delta)\setminus\mathcal{G}'(s,\delta) \colon\mathcal{P}\cap\Lambda^\ast\ne \{\mathbf 0\} \\ &\qquad \implies \ \ \exists\mathcal{P}\in\mathcal{F}'(t,\gamma):(d-1)\mathcal{P} \cap\Lambda\ne \{\mathbf 0\}. \end{aligned}
\end{equation}
\tag{124}
$$
Clearly, (124) and (113) give the required implication (114). Thus, in view of (106), Theorem 33 is proved. 5.2.4. A refinement of Theorem 34We actually proved something stronger than Theorem 34 in § 5.2.3. Implication (120) can naturally be generalised to the case of an arbitrary parallelepiped $\mathcal{P}(\boldsymbol{\lambda}^\ast,\boldsymbol{\mu}^\ast) \in\mathcal{G}(s,\delta)$. First, we note that for $\mathcal{P}(\boldsymbol{\lambda}^\ast,\boldsymbol{\mu}^\ast) \in\mathcal{G}'(s,\delta)$ we can set $k$ to be equal to zero, since by (107) we have $\lambda_j\geqslant1$ for each $j$. In this case the equality $\widehat{\boldsymbol{\lambda}}=\boldsymbol{\lambda}$ is an analogue of (118). Second, the definition of $\widehat{\boldsymbol{\lambda}}$ is naturally generalised to the case of an arbitrary ordering of the components of $\boldsymbol{\lambda}$. Let $\lambda_{j_1}\leqslant\cdots\leqslant\lambda_{j_m}$. If $\lambda_{j_1}<1$, then we set $k$ to be equal to the greatest index such that $\lambda_{j_1}\cdots\lambda_{j_k}<1$. If $\lambda_{j_1}\geqslant1$, then we set ${k=0}$. Finally, we define the tuple $\widehat{\boldsymbol{\lambda}} =(\widehat\lambda_1,\dots,\widehat\lambda_m)$ by
$$
\begin{equation}
\begin{aligned} \, \widehat\lambda_{j_i}&=1,&\qquad i&=1,\dots,k, \\ \widehat\lambda_{j_i}&= \lambda_{j_i}(\lambda_{j_1}\cdots\lambda_{j_k})^{1/(m-k)},&\qquad i&=k+1,\dots,m. \end{aligned}
\end{equation}
\tag{125}
$$
Arguing in the manner of § 5.2.3, we arrive at the following refinement of Theorem 34. Theorem 35. Let $\Lambda$ and $\Lambda^\ast$ be defined by (100). Given arbitrary tuples $\boldsymbol{\lambda}=(\lambda_1,\dots,\lambda_m)\in\mathbb{R}_+^m$ and $\boldsymbol{\mu}=(\mu_1,\dots,\mu_n)\in\mathbb{R}_+^n$, define $\boldsymbol{\lambda}^\ast$ and $\boldsymbol{\mu}^\ast$ by (107) and $\widehat{\boldsymbol{\lambda}}$ by (125). Then Moreover, $\pi(\widehat{\boldsymbol{\lambda}},\boldsymbol{\mu}) =\pi(\boldsymbol{\lambda},\boldsymbol{\mu})$, $\min_{1\leqslant j\leqslant m}\widehat\lambda_j\geqslant1$, and Theorem 35 is the ‘core’ of the multiplicative transference principle, just as Theorem 22 is the ‘core’ of Khintchine’s transference principle. We note that, under the assumption that $\mathcal{P}(\boldsymbol{\lambda}^\ast,\boldsymbol{\mu}^\ast) \cap\Lambda^\ast\ne\{\mathbf{0}\}$, Theorem 22 guarantees the existence of non-zero points of $\Lambda$ in the parallelepiped $(d-1)\mathcal{P}(\boldsymbol{\lambda},\boldsymbol{\mu})$, which is not contained in $(d-1)\mathcal{P}(\widehat{\boldsymbol{\lambda}},\boldsymbol{\mu})$ in the case when $\boldsymbol{\lambda}$ has components strictly smaller than $1$ as well as ones strictly greater than $1$. It turns out that there is a whole family of pairwise distinct parallelepipeds, containing each a non-zero point of $\Lambda$. The exact formulation of this fact can be found in the paper [62] devoted to improving Mahler’s theorem. We note also that the constant $d-1$ in (126) can be replaced by a smaller one, which tends to $1$ as $d\to\infty$; for instance, by $d^{1/(2d-2)}$ — the same constant as in relation (46) strengthening the statement of Theorem 22. Details can be found in [77]. 5.3. Multiplicatively badly approximable matricesThe multiplicative setting also admits talking about badly approximable matrices. Definition 23. A matrix $\Theta$ is called multiplicatively badly approximable if there is a positive constant $c$ depending only on $\Theta$ such that for each pair $(\mathbf{x},\mathbf{y})\in\mathbb{Z}^m\oplus\mathbb{Z}^n$ with non-zero $\mathbf{x}$ we have While the existence of badly approximable matrices can be proved rather easily, the question whether there exist multiplicatively badly approximable matrices is open. Even in the simplest case $n=2$, $m=1$ the statement that multiplicatively badly approximable matrices exist is precisely the negation of Littlewood’s conjecture (its formulation can be found in the beginning of § 5). In 1955 Cassels and Swinnerton-Dyer [78] proved that if $n=2$, $m=1$ and $\Theta$ is multiplicatively badly approximable, then so is $\Theta^\top$. Theorem 35 enables one to prove the converse statement as well as its analogue for an arbitrary matrix. Indeed, by analogy with the deduction of (106), it can be shown that $\Theta$ is multiplicatively badly approximable if and only if there is a positive constant $c$ such that no parallelepiped in the family
$$
\begin{equation*}
\Bigl\{\mathcal{P}(\boldsymbol{\lambda},\boldsymbol{\mu}) \bigm| \pi(\boldsymbol{\lambda},\boldsymbol{\mu})=c,\ \min_{1\leqslant j\leqslant m}\lambda_j\geqslant1 \Bigr\}
\end{equation*}
\notag
$$
contains non-zero points of $\Lambda$ (for the notation, see (100), (101), and (109)). In a similar way, $\Theta^\top$ is multiplicatively badly approximable if and only if there is a positive constant $c$ such that no parallelepiped in the family
$$
\begin{equation*}
\Bigl\{\mathcal{P}(\boldsymbol{\lambda},\boldsymbol{\mu}) \bigm| \pi(\boldsymbol{\lambda},\boldsymbol{\mu})=c,\ \min_{1\leqslant i\leqslant n}\mu_i\geqslant1 \Bigr\}
\end{equation*}
\notag
$$
contains non-zero points of $\Lambda^\ast$. Since $\pi(\boldsymbol{\lambda}^\ast,\boldsymbol{\mu}^\ast) =\pi(\boldsymbol{\lambda},\boldsymbol{\mu})^{d-1}$, it follows immediately from Theorem 35 that if $\Theta^\top$ is not multiplicatively badly approximable, then neither is $\Theta$. Thus, the following theorem is valid. Theorem 36. A matrix $\Theta$ is multiplicatively badly approximable if and only if so is $\Theta^\top$. 6. Diophantine approximation with weightsAs before, let us consider a matrix
$$
\begin{equation*}
\Theta=\begin{pmatrix} \theta_{11} & \dots & \theta_{1m} \\ \vdots & \ddots & \vdots \\ \theta_{n1} & \dots & \theta_{nm} \end{pmatrix},\qquad \theta_{ij}\in\mathbb{R},\quad m+n\geqslant3,
\end{equation*}
\notag
$$
and the system of linear equations with the variables $\mathbf{x}=(x_1,\dots,x_m)\in\mathbb{R}^m$ and $\mathbf{y}=(y_1,\dots,y_n)\in\mathbb{R}^n$. In previous sections we observed two approaches in measuring the ‘magnitude’ of $\Theta\mathbf{x}-\mathbf{y}$: the first uses the sup-norm, and the second uses the geometric mean of the absolute values of coordinates. There is also an ‘intermediate’ approach, so-called Diophantine approximation with weights. Let us fix weights
$$
\begin{equation*}
\begin{gathered} \, \boldsymbol{\sigma}=(\sigma_1,\dots,\sigma_m)\in\mathbb{R}_{>0}^m\quad\text{and}\quad \boldsymbol{\rho}=(\rho_1,\dots,\rho_n)\in\mathbb{R}_{>0}^n, \\ \sigma_1\geqslant\cdots\geqslant\sigma_m,\qquad \rho_1\geqslant\cdots\geqslant\rho_n,\qquad \sum_{j=1}^m\sigma_j=\sum_{i=1}^n\rho_i=1, \end{gathered}
\end{equation*}
\notag
$$
and define the weighted norms $|\,{\cdot}\,|_{\boldsymbol{\sigma}}$ and $|\,{\cdot}\,|_{\boldsymbol{\rho}}$ by
$$
\begin{equation*}
|\mathbf x|_{\boldsymbol{\sigma}}=\max_{1\leqslant j\leqslant m} |x_j|^{1/\sigma_j}\quad\text{for}\ \mathbf x=(x_1,\dots,x_m)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
|\mathbf y|_{\boldsymbol{\rho}}=\max_{1\leqslant i\leqslant n} |y_i|^{1/\rho_i}\quad\text{for}\ \mathbf y=(y_1,\dots,y_n).
\end{equation*}
\notag
$$
Consider the system of inequalities
$$
\begin{equation}
\begin{cases} |\mathbf x|_{\boldsymbol{\sigma}}\leqslant t, \\ |\Theta\mathbf x-\mathbf y|_{\boldsymbol{\rho}}\leqslant t^{-\gamma}. \end{cases}
\end{equation}
\tag{127}
$$
Definition 24. The supremum of real $\gamma$ satisfying the condition that there exist arbitrarily large $t$ such that (respectively, for every $t$ large enough) the system (127) admits a non-zero solution $(\mathbf{x},\mathbf{y})\in\mathbb{Z}^m\oplus\mathbb{Z}^n$ is called the regular (respectively, uniform) weighted Diophantine exponent of $\Theta$ and is denoted by $\omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta)$ (respectively, $\widehat\omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta)$). It is easily verified that in the case of ‘trivial’ weights, that is, when all the $\sigma_j$ are equal to $1/m$ and all the $\rho_i$ are equal to $1/n$, we are dealing with ordinary Diophantine exponents, as in this case we have
$$
\begin{equation*}
\omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta)= \frac{n}{m}\,\omega(\Theta)\quad\text{and}\quad \widehat\omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta)= \frac{n}{m}\,\widehat\omega(\Theta).
\end{equation*}
\notag
$$
Minkowski’s convex body theorem applied to system (127) yields the ‘trivial’ inequalities
$$
\begin{equation*}
\omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta)\geqslant \widehat\omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta)\geqslant1,
\end{equation*}
\notag
$$
analogous to (74), which hold for every choice of $\boldsymbol{\sigma}$ and $\boldsymbol{\rho}$. 6.1. Schmidt’s Bad-conjectureIn the case $m=1$, $n=2$ we can observe a very close relationship between Diophantine approximation with weights and Littlewood’s conjecture, which relates to multiplicative Diophantine approximation (the formulation of this conjecture is presented in the beginning of § 5). As we noted in § 5.3, the negation of this conjecture is equivalent to the statement that for $m=1$, $n=2$ multiplicatively badly approximable matrices exist. Definition 25. A matrix $\Theta$ is called badly approximable with weights $\boldsymbol{\sigma}$ and $\boldsymbol{\rho}$ if there is a positive constant $c$ depending only on $\Theta$ such that for each pair $(\mathbf{x},\mathbf{y})\in\mathbb{Z}^m\oplus\mathbb{Z}^n$ with non-zero $\mathbf{x}$ we have In [65] Schmidt noted that Davenport’s construction from [79] can easily be modified to prove that for $m=1$, $n=2$ there exist badly approximable matrices $\Theta$ with arbitrary weights $\boldsymbol{\rho}=(\rho_1,\rho_2)$ and conjectured that for any two distinct sets of weights $\boldsymbol{\rho}'$ and $\boldsymbol{\rho}''$ there exist matrices that are simultaneously badly approximable with weights $\boldsymbol{\rho}'$ and with weights $\boldsymbol{\rho}''$. It is easy to see that the existence of a counterexample to this conjecture, that is, the existence of two sets of weights $\boldsymbol{\rho}'$ and $\boldsymbol{\rho}''$ such that any matrix $\Theta\in\mathbb{R}^{2\times1}$ is not badly approximable with at least one of them, implies Littlewood’s conjecture. In 2011 Badziahin, Pollington, and Velani published the paper [80], where they proved Schmidt’s conjecture formulated above. Note that not only did they prove the existence of matrices $\Theta\in\mathbb{R}^{2\times1}$ that are badly approximable with weights $\boldsymbol{\rho}'$ and with weights $\boldsymbol{\rho}''$, but they also proved that there are pretty many such matrices. More specifically, they proved that, given $k$ sets of weights $\boldsymbol{\rho}^{(1)},\dots,\boldsymbol{\rho}^{(k)}$, the Hausdorff dimension of the set of matrices that are badly approximable with each of these $k$ sets of weights is $2$. 6.2. Transference theoremsThe first transference inequality for Diophantine approximation with weights generalising Dyson’s inequality (75) was obtained in [81]. In that paper the authors showed that
$$
\begin{equation}
\omega_{\boldsymbol{\rho},\boldsymbol{\sigma}}(\Theta^\top)\geqslant \frac{(m+n-1)(\rho_n^{-1}+\sigma_m^{-1}\omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta))+ \sigma_1^{-1}(\omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta)-1)} {(m+n-1)(\rho_n^{-1}+\sigma_m^{-1}\omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta))- \rho_1^{-1}(\omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta)-1)}\,.
\end{equation}
\tag{128}
$$
As it turned out later, inequality (128) is not optimal. It was improved in [64]. Theorem 37 (German, 2020). For every matrix $\Theta\in\mathbb{R}^{n\times m}$ and arbitrary weights $\boldsymbol{\sigma}$ and $\boldsymbol{\rho}$ we have In the same paper [64] a transference theorem for uniform weighted exponents was proved. It generalises Theorem 29. Theorem 38 (German, 2020). For every matrix $\Theta\in\mathbb{R}^{n\times m}$, $m+n\geqslant3$, and arbitrary weights $\boldsymbol{\sigma}$ and $\boldsymbol{\rho}$ we have As before, if some denominator happens to be equal to zero, then we assume the corresponding expression to take the value $+\infty$. 6.2.1. Embedding into $\mathbb{R}^{m+n}$We can prove Theorems 37 and 38 by following the scheme we discussed in § 4.4. As before, set and consider the same lattices
$$
\begin{equation}
\Lambda=\Lambda(\Theta)=\begin{pmatrix} \mathbf I_m & \\ -\Theta & \mathbf I_n \end{pmatrix}\mathbb{Z}^d\quad\text{and}\quad \Lambda^\ast=\Lambda^\ast(\Theta)= \begin{pmatrix} \mathbf I_m & \Theta^\top \\ & \mathbf I_n \end{pmatrix} \mathbb{Z}^d.
\end{equation}
\tag{131}
$$
Instead of the parallelepipeds (86) and (87), consider the parallelepipeds
$$
\begin{equation}
\mathcal{P}(t,\gamma) =\biggl\{\,\mathbf z= (z_1,\dots,z_d)\in\mathbb{R}^d \biggm| \begin{aligned} \, &|(z_1,\dots,z_m)|_{\boldsymbol{\sigma}}\leqslant t \\ &|(z_{m+1},\dots,z_d)|_{\boldsymbol{\rho}}\leqslant t^{-\gamma} \end{aligned} \biggr\}
\end{equation}
\tag{132}
$$
and
$$
\begin{equation}
\mathcal{Q}(s,\delta) =\biggl\{\,\mathbf z=(z_1,\dots,z_d)\in \mathbb{R}^d \biggm| \begin{aligned} \, &|(z_1,\dots,z_m)|_{\boldsymbol{\sigma}}\leqslant s^{-\delta} \\ &|(z_{m+1},\dots,z_d)|_{\boldsymbol{\rho}}\leqslant s \end{aligned} \biggr\}.
\end{equation}
\tag{133}
$$
Then
$$
\begin{equation*}
\begin{aligned} \, \omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta) & = \sup\bigl\{\gamma\geqslant1 \bigm| \forall\,t_0\in\mathbb{R}\,\ \exists\,t>t_0\colon \mathcal{P}(t,\gamma)\cap\Lambda\ne \{\mathbf 0\} \bigr\}, \\ \widehat\omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta) & = \sup\bigl\{\gamma\geqslant1 \bigm| \exists\,t_0\in\mathbb{R}\colon \forall\,t>t_0 \ \mathcal{P}(t,\gamma)\cap\Lambda\ne \{\mathbf 0\}\bigr\}, \\ \omega_{\boldsymbol{\rho},\boldsymbol{\sigma}}(\Theta^\top) & = \sup\bigl\{\delta\geqslant1 \bigm| \forall\,s_0\in\mathbb{R}\,\ \exists\,s>s_0\colon \mathcal{Q}(s,\delta)\cap\Lambda^\ast\ne \{\mathbf 0\}\bigr\}, \end{aligned}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\widehat\omega_{\boldsymbol{\rho},\boldsymbol{\sigma}}(\Theta^\top) = \sup\bigl\{\delta\geqslant1 \bigm| \exists\,s_0\in\mathbb{R}\colon \forall\,s>s_0\ \mathcal{Q}(s,\delta)\cap\Lambda^\ast\ne \{\mathbf 0\}\bigr\}.
\end{equation*}
\notag
$$
6.2.2. Deriving the theorem on regular exponentsFor $s>1$ and $\delta\geqslant1$ set
$$
\begin{equation}
t=s^{(\sigma_m^{-1}+(\rho_n^{-1}-1)\delta)/(\sigma_m^{-1}+ \rho_n^{-1}-1)}\quad\text{and}\quad \gamma=\frac{(\sigma_m^{-1}-1)+\rho_n^{-1}\delta} {\sigma_m^{-1}+(\rho_n^{-1}-1)\delta}\,.
\end{equation}
\tag{134}
$$
Such a correspondence provides the inequalities
$$
\begin{equation*}
s^{-\delta\sigma_j} \leqslant t^{-\sigma_j+1-\gamma},\qquad j=1,\dots,m,
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
s^{\rho_i} \leqslant t^{\gamma\rho_i+1-\gamma},\qquad i=1,\dots,n,
\end{equation*}
\notag
$$
which imply that
$$
\begin{equation}
\mathcal{Q}(s,\delta)\subseteq \mathcal{P}(t,\gamma)^\ast,
\end{equation}
\tag{135}
$$
since, according to Definition 14 (see § 3.4.2), the compound of $\mathcal{P}(t,\gamma)$ has the form
$$
\begin{equation*}
\mathcal{P}(t,\gamma)^\ast=\biggl\{\mathbf z=(z_1,\dots,z_d)\in \mathbb{R}^d \biggm| \begin{alignedat}{2} &|z_j|\leqslant t^{-\sigma_j+1-\gamma}, &\quad j&=1,\dots,m \\ &|z_{m+i}|\leqslant t^{\gamma\rho_i+1-\gamma}, &\quad i&=1,\dots,n \end{alignedat} \biggr\}.
\end{equation*}
\notag
$$
Applying Mahler’s theorem in disguise of Theorem 22 again we obtain
$$
\begin{equation}
\mathcal{Q}(s,\delta)\cap\Lambda^\ast\ne \{\mathbf 0\}\ \ \implies\ \ (d-1)\mathcal{P}(t,\gamma)\cap\Lambda\ne \{\mathbf 0\}.
\end{equation}
\tag{136}
$$
Thus,
$$
\begin{equation*}
\omega_{\boldsymbol{\rho},\boldsymbol{\sigma}}(\Theta^\top)\geqslant\delta \ \ \implies\ \ \omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta)\geqslant\gamma= \frac{(\sigma_m^{-1}-1)+\rho_n^{-1}\delta} {\sigma_m^{-1}+(\rho_n^{-1}-1)\delta}\,,
\end{equation*}
\notag
$$
so that
$$
\begin{equation*}
\omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta)\geqslant \frac{(\sigma_m^{-1}-1)+\rho_n^{-1}\omega_{\boldsymbol{\rho},\boldsymbol{\sigma}}(\Theta^\top)} {\sigma_m^{-1}+(\rho_n^{-1}-1)\omega_{\boldsymbol{\rho},\boldsymbol{\sigma}}(\Theta^\top)}\,.
\end{equation*}
\notag
$$
Swapping the triple $(\boldsymbol{\sigma},\boldsymbol{\rho},\Theta)$ for $(\boldsymbol{\rho},\boldsymbol{\sigma},\Theta^\top)$, we obtain (129). 6.2.3. Deriving the theorem on uniform exponentsWe apply the construction of ‘nodes’ and ‘leaves’ described in § 3.4.5. Instead of the parallelepipeds $\mathcal{Q}_r$ defined by (55), let us consider the parallelepipeds
$$
\begin{equation*}
\mathcal{Q}_r=\bigg\{\,\mathbf z=(z_1,\dots,z_d) \in\mathbb{R}^d \biggm| \begin{aligned} \, &|(z_1,\dots,z_m)|_{\boldsymbol{\sigma}}\leqslant (hH/r)^{-\alpha} \\ &|(z_{m+1},\dots,z_d)|_{\boldsymbol{\rho}}\leqslant r \end{aligned} \bigg\}.
\end{equation*}
\notag
$$
Correspondingly, instead of (47) and (48) we define parallelepipeds $\mathcal{P}(t,\gamma)$ and $\mathcal{Q}(s,\delta)$ by (132) and (133), respectively. Let us use Fig. 8 with the agreement that now $u$ and $v$ denote $|(z_{m+1},\dots,z_d)|_{\boldsymbol{\rho}}$ and $|(z_1,\dots,z_m)|_{\boldsymbol{\sigma}}$, respectively. Upon fixing $s>1$ and $\delta\geqslant1$, we define $t$ and $\gamma$ by (134) and set
$$
\begin{equation*}
h=s,\qquad \beta=\delta,\quad\text{and}\quad \alpha=\begin{cases} \dfrac{\sigma_m^{-1}+(\rho_n^{-1}-1)\delta}{\sigma_m^{-1}+\rho_n^{-1}-1} & \text{if}\ \ \delta\geqslant\dfrac{\rho_n(\sigma_m^{-1}-1)} {\sigma_m(\rho_n^{-1}-1)}\,, \\ \dfrac{(\sigma_m^{-1}+\rho_n^{-1}-1)\delta}{(\sigma_m^{-1}-1)+ \rho_n^{-1}\delta} & \text{if}\ \ \delta\leqslant\dfrac{\rho_n(\sigma_m^{-1}-1)} {\sigma_m(\rho_n^{-1}-1)}\,. \end{cases}
\end{equation*}
\notag
$$
For such a choice of parameters the quantities $\gamma$ and $\alpha$ are related by the equality
$$
\begin{equation}
\gamma=\begin{cases} \dfrac{1-\rho_n\alpha^{-1}}{1-\rho_n} & \text{if}\ \ \alpha\geqslant\dfrac{\rho_n}{\sigma_m}\,, \\ \dfrac{1-\sigma_m}{1-\sigma_m\alpha} & \text{if}\ \ \alpha\leqslant\dfrac{\rho_n}{\sigma_m}\,. \end{cases}
\end{equation}
\tag{137}
$$
Arguing in the manner of § 3.4.6, we obtain the implication
$$
\begin{equation*}
\widehat\omega_{\boldsymbol{\rho},\boldsymbol{\sigma}}(\Theta^\top)\geqslant\alpha\ \ \implies\ \ \widehat\omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta)\geqslant\gamma,
\end{equation*}
\notag
$$
so that
$$
\begin{equation*}
\widehat\omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta)\geqslant \begin{cases} \dfrac{1-\rho_n\widehat\omega_{\boldsymbol{\rho},\boldsymbol{\sigma}}(\Theta^\top)^{-1}} {1-\rho_n} & \text{if}\ \ \widehat\omega_{\boldsymbol{\rho},\boldsymbol{\sigma}} (\Theta^\top)\geqslant\dfrac{\rho_n}{\sigma_m}\,, \\ \dfrac{1-\sigma_m\vphantom{1^{\textstyle|}}} {1-\sigma_m\widehat\omega_{\boldsymbol{\rho},\boldsymbol{\sigma}}(\Theta^\top)} & \text{if}\ \ \widehat\omega_{\boldsymbol{\rho},\boldsymbol{\sigma}}(\Theta^\top) \leqslant\dfrac{\rho_n}{\sigma_m}\,. \end{cases}
\end{equation*}
\notag
$$
Swapping the triple $(\boldsymbol{\sigma},\boldsymbol{\rho},\Theta)$ for $(\boldsymbol{\rho},\boldsymbol{\sigma},\Theta^\top)$, we arrive at (130). 6.3. Marnat’s resultFor $m=1$ and $n=2$ (that is, in the most interesting case for Littlewood’s conjecture) we have $\sigma_1=1$, $\rho_n=\rho_2$, and $1-\rho_n=\rho_1$, which simplifies the appearance of inequality (130) significantly. Swapping $(\boldsymbol{\sigma},\boldsymbol{\rho},\Theta)$ for $(\boldsymbol{\rho},\boldsymbol{\sigma},\Theta^\top)$, as usual, gives one more inequality. Moreover, in this case, just as in the problem of classical simultaneous approximation (see inequality (16) in § 3.2), we have the inequality $\widehat\omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta) \leqslant\rho_1^{-1}$ for the uniform exponent (under the assumption that not all the components of $\Theta$ are rational). Therefore, Theorem 38 yields the following statement. Corollary 12. Let $m=1$ and $n=2$, and let $\Theta\notin\mathbb{Q}^{2\times1}$. Then In [82] Marnat proved that for every positive $b<(3\rho_1)^{-1}$ and every $a$ satisfying the inequalities
$$
\begin{equation*}
\begin{aligned} \, \rho_1a+\rho_2b&\geqslant1, \\ \rho_2a+\rho_1b&\leqslant1, \\ \rho_1a&\leqslant1, \end{aligned}
\end{equation*}
\notag
$$
there exist continuum many matrices $\Theta\in\mathbb{R}^{2\times1}$ such that $\widehat\omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta)=a$ and $\widehat\omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta^\top)=b^{-1}$. It follows that, if $\widehat\omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta^\top)>3\rho_1$ and $\theta_{11}$ is irrational, inequalities (138), and (139) are sharp. In particular, this result of Marnat implies that even for $m+n=3$ there is no analogue of Jarník’s inequality (20) in the case of non-trivial weights. 6.4. Parametric geometry of numbersThe approach described in §§ 3.4.8 and 4.4.4 is also applicable to Diophantine approximation with weights. We define the lattices $\Lambda$ and $\Lambda^\ast$ by (131), just as in § 4.4.4. As for the subspace of the space of parameters $\mathcal{T}$ corresponding to the problem of Diophantine approximation with non-trivial weights, unlike in the case of trivial weights, we need more than just a single one-dimensional subspace. Given weights $\boldsymbol{\sigma}=(\sigma_1,\dots,\sigma_m)$ and $\boldsymbol{\rho}=(\rho_1,\dots,\rho_n)$, set
$$
\begin{equation*}
\mathbf e_1=\mathbf e_1(\boldsymbol{\sigma},\boldsymbol{\rho})= (1-d\sigma_1,\dots,1-d\sigma_m,\underbrace{1,\dots,1}_{n}\,)
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\mathbf e_2=\mathbf e_2(\boldsymbol{\sigma},\boldsymbol{\rho})= (\underbrace{1,\dots,1}_{m},1-d\rho_n,\dots,1-d\rho_1).
\end{equation*}
\notag
$$
If every $\sigma_j$ is equal to $1/m$ and every $\rho_i$ is equal to $1/n$, then the vectors $\mathbf{e}_1$ and $\mathbf{e}_2$ are proportional. But if the weights are non-trivial, then $\mathbf{e}_1$ and $\mathbf{e}_2$ span a two-dimensional subspace of $\mathcal{T}$. For each $\gamma,\delta\in\mathbb{R}$ set
$$
\begin{equation*}
\boldsymbol{\mu}_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\gamma)=\gamma\mathbf e_2-\mathbf e_1
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\boldsymbol{\mu}^\ast_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\delta)=\delta\mathbf e_1-\mathbf e_2.
\end{equation*}
\notag
$$
Every $\boldsymbol{\mu}_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\gamma)$ and every $\boldsymbol{\mu}^\ast_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\delta)$ determine one-dimensional subspaces of $\mathcal{T}$. Correspondingly, we consider the family of paths $\mathfrak{T}_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\gamma)$ determined by the mappings $s\mapsto s\boldsymbol{\mu}_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\gamma)$ and the family of paths $\mathfrak{T}^\ast_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\delta)$ determined by the mappings $s\mapsto s\boldsymbol{\mu}^\ast_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\delta)$. We have lower and upper Schmidt–Summerer exponents of the first and second types defined for these paths (see Definition 17). It was shown in [68] that weighted Diophantine exponents are related to Schmidt–Summerer exponents as follows:
$$
\begin{equation}
\begin{aligned} \, \omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta)&\geqslant\gamma \ \ \iff \ \ \underline{\varphi}_1\bigl(\Lambda,\mathfrak{T}_{\boldsymbol{\sigma},\boldsymbol{\rho}} (\gamma)\bigr)\leqslant1-\gamma, \\ \omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta)&\leqslant\gamma \ \ \iff \ \ \underline{\varphi}_1\bigl(\Lambda,\mathfrak{T}_{\boldsymbol{\sigma},\boldsymbol{\rho}} (\gamma)\bigr)\geqslant1-\gamma, \\ \widehat\omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta)&\geqslant\gamma \ \ \iff \ \ \overline{\varphi}_1\bigl(\Lambda,\mathfrak{T}_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\gamma)\bigr) \leqslant1-\gamma, \\ \widehat\omega_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\Theta)&\leqslant\gamma \ \ \iff \ \ \overline{\varphi}_1\bigl(\Lambda,\mathfrak{T}_{\boldsymbol{\sigma},\boldsymbol{\rho}}(\gamma)\bigr) \geqslant1-\gamma \end{aligned}
\end{equation}
\tag{140}
$$
and, in a similar way,
$$
\begin{equation}
\begin{aligned} \, \omega_{\boldsymbol{\rho},\boldsymbol{\sigma}}(\Theta^\top)&\geqslant\delta \ \ \iff \ \ \underline{\varphi}_1\bigl(\Lambda^\ast,\mathfrak{T}^\ast_{\boldsymbol{\sigma},\boldsymbol{\rho}} (\delta)\bigr)\leqslant1-\delta, \\ \omega_{\boldsymbol{\rho},\boldsymbol{\sigma}}(\Theta^\top)&\leqslant\delta \ \ \iff \ \ \underline{\varphi}_1\bigl(\Lambda^\ast,\mathfrak{T}^\ast_{\boldsymbol{\sigma},\boldsymbol{\rho}} (\delta)\bigr)\geqslant1-\delta, \\ \widehat\omega_{\boldsymbol{\rho},\boldsymbol{\sigma}}(\Theta^\top)&\geqslant\delta \ \ \iff \ \ \overline{\varphi}_1\bigl(\Lambda^\ast,\mathfrak{T}^\ast_{\boldsymbol{\sigma},\boldsymbol{\rho}} (\delta)\bigr)\leqslant1-\delta, \\ \widehat\omega_{\boldsymbol{\rho},\boldsymbol{\sigma}}(\Theta^\top)&\leqslant\delta \ \ \iff \ \ \overline{\varphi}_1\bigl(\Lambda^\ast,\mathfrak{T}^\ast_{\boldsymbol{\sigma},\boldsymbol{\rho}} (\delta)\bigr)\geqslant1-\delta. \end{aligned}
\end{equation}
\tag{141}
$$
It was shown in the same paper that if $\delta\geqslant1$ and $\gamma$ is related to $\delta$ by the right-hand equality in (134), then
$$
\begin{equation}
\underline{\varphi}_1\bigl(\Lambda^\ast,\mathfrak{T}^\ast_{\boldsymbol{\sigma},\boldsymbol{\rho}} (\delta)\bigr)\leqslant 1-\delta\ \ \implies\ \ \underline{\varphi}_1\bigl(\Lambda,\mathfrak{T}_{\boldsymbol{\sigma},\boldsymbol{\rho}} (\gamma)\bigr)\leqslant 1-\gamma.
\end{equation}
\tag{142}
$$
In view of (140) and (141) the implication (142) is nothing else but inequality (129) written in terms of Schmidt–Summerer exponents. Like with Khintchine’s and Dyson’s inequalities, this approach enables splitting inequality (129) into a chain of inequalities between intermediate exponents. The corresponding definitions and formulations can be found in [68].
7. Diophantine exponents of latticesIn the previous sections we considered a matrix $\Theta\in\mathbb{R}^{n\times m}$ and dealt with the question of how rapidly the vector $\Theta\mathbf{x}-\mathbf{y}$ can tend to zero if $\mathbf{x}$ and $\mathbf{y}$ are assumed to range over integer vectors. In other words, we considered $n$ linear forms with coefficients written in the rows of the matrix
$$
\begin{equation*}
\begin{pmatrix} \Theta & -\mathbf I_n \end{pmatrix}=\begin{pmatrix} \theta_{11} & \dots & \theta_{1m} & -1 & \dots & 0 \\ \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ \theta_{n1} & \dots & \theta_{nm} & 0 & \dots & -1 \end{pmatrix}.
\end{equation*}
\notag
$$
and examined their values at integer points. We considered several ways to measure the ‘magnitude’ of a vector. The first uses the sup-norm, the second uses the geometric mean of the absolute values of coordinates, and the third uses weighted norms. But each time we worked with $n$ linear forms in $d=m+n$ variables. That is, every time the number of linear forms was strictly less than the dimension of the ambient space. This section is devoted to problems concerning $d$-tuples of linear forms in $d$ variables. Let $L_1,\dots,L_d$ be linearly independent linear forms in $d$ variables. Consider the lattice
$$
\begin{equation*}
\Lambda=\{(L_1(\mathbf u),\dots,L_d(\mathbf u))\mid \mathbf u\in\mathbb{Z}^d\}.
\end{equation*}
\notag
$$
Using a norm to measure the ‘magnitude’ of elements of $\Lambda$ is not very productive, as such a norm is bounded away from zero at non-zero points of $\Lambda$. But the geometric mean of the absolute values of coordinates leads to very interesting and sometimes very difficult problems. As in § 5, for each $\mathbf{z}=(z_1,\dots,z_d)\in\mathbb{R}^d$ set
$$
\begin{equation*}
\Pi(\mathbf z)=\prod_{1\leqslant i\leqslant d}|z_i|^{1/d}.
\end{equation*}
\notag
$$
We also recall that $|\,{\cdot}\,|$ denotes the sup-norm. Definition 26. The supremum of real $\gamma$ such that the inequality admits infinitely many solutions in $\mathbf{z}\in\Lambda$ is called the Diophantine exponent of $\Lambda$ and is denoted by $\omega(\Lambda)$. 7.1. The spectrum of lattice exponentsIt follows from Minkowski’s convex body theorem that for every lattice $\Lambda$ the ‘trivial’ inequality holds. It is clear that $\omega(\Lambda)=0$ whenever the functional $\Pi(\mathbf{x})$ is bounded away from zero at the non-zero points of $\Lambda$. For instance, this is the case if $\Lambda$ is the lattice of a complete module in a totally real algebraic extension of $\mathbb{Q}$, that is, if
$$
\begin{equation}
\Lambda=\begin{pmatrix} \sigma_1(\omega_1) & \sigma_1(\omega_2) & \dots & \sigma_1(\omega_d) \\ \sigma_2(\omega_1) & \sigma_2(\omega_2) & \dots & \sigma_2(\omega_d) \\ \vdots & \vdots & \ddots & \vdots \\ \sigma_d(\omega_1) & \sigma_d(\omega_2) & \dots & \sigma_d(\omega_d) \end{pmatrix}\mathbb{Z}^d,
\end{equation}
\tag{143}
$$
where $\omega_1,\dots,\omega_d$ is a basis of a totally real extension $E$ of $\mathbb{Q}$ of degree $d$ and $\sigma_1,\dots,\sigma_d$ are the embeddings of $E$ into $\mathbb{R}$. Such lattices are often called algebraic. A detailed account on algebraic lattices can be found, for instance, in the book [83] by Borevich and Shafarevich. There is a wider class of lattices with $\omega(\Lambda)=0$. Their existence is provided by Schmidt’s famous subspace theorem, which was published by Schmidt [84] in 1972. Its formulation and proof can also be found in the book [85] by Bombieri and Gubler. It was shown in [86] and [87] that if the coefficients of linear forms $L_1,\dots,L_d$ are algebraic and for every $k$-tuple $(i_1,\dots,i_k)$, $1\leqslant i_1<\dots<i_k\leqslant d$, $1\leqslant k\leqslant d$, the coefficients of the multivector $L_{i_1}\wedge\dots\wedge L_{i_k}$ are linearly independent over $\mathbb{Q}$, then $\omega(\Lambda)=0$ by the subspace theorem. It was shown in [87] that by weakening the condition of linear independence formulated above examples for $d\geqslant3$ of lattices with exponents attaining values
$$
\begin{equation}
\frac{ab}{cd}\,,\qquad a,b,c\in\mathbb{N},\quad a+b+c=d,
\end{equation}
\tag{144}
$$
can be constructed. It is natural to conjecture that there exist lattices $\Lambda$ with any prescribed non- negative $\omega(\Lambda)$. For $d=2$ this is easily proved with the help of continued fractions (see § 2.5). For $d\geqslant3$ this question is still open. It was proved in [88] that the interval is contained in the spectrum of the values of $\omega(\Lambda)$. The question whether there are positive numbers in this spectrum that are not in the interval (145) and are not equal to any of the numbers (144) is still open for ${d\geqslant3}$. 7.2. Lattices with positive norm minimumFor lattices, an analogue of the property of a number to be badly approximable is the property to have a positive norm minimum. Definition 27. Let $\Lambda$ be a full-rank lattice in $\mathbb{R}^d$. Then its norm minimum is defined by As we said in the previous section, if $\Lambda$ is an algebraic lattice, then $N(\Lambda)>0$. Cassels and Swinnerton-Dyer [78] showed that Littlewood’s conjecture (see the formulation in the beginning of § 5) can be derived from the following hypothesis, which is an analogue of Oppenheim’s conjecture on quadratic forms for decomposable forms of degree $d$. The latter was proved by Margulis in the mid-1980s (see [89]). Cassels–Swinnerton-Dyer’s Conjecture. Let $d\geqslant3$ and let $\Lambda$ be a full-rank lattice in $\mathbb{R}^d$. Then the condition $N(\Lambda)>0$ is equivalent to the existence of a non-degenerate diagonal operator $D$ such that $D\Lambda$ is an algebraic lattice, that is, satisfies (143). The main tool in the proof of the fact that Littlewood’s conjecture follows from Cassels–Swinnerton-Dyer’s conjecture is Mahler’s compactness criterion (see [10] and [78]). For $d=2$ the statement of Cassels–Swinnerton-Dyer’s conjecture is not true. Indeed, in view of Proposition 3 and the geometric interpretation of continued fractions described in § 2.5, in the two-dimensional case the lattice
$$
\begin{equation*}
\Lambda=\{(L_1(\mathbf u),L_2(\mathbf u))\mid\mathbf u\in\mathbb{Z}^2\}
\end{equation*}
\notag
$$
has a positive norm minimum if and only if the ratio of the coefficients of $L_1$ and the ratio of the coefficients of $L_2$ are badly approximable numbers. Thus, if these two ratios are irrational numbers with bounded partial quotients, but the sequences of these partial quotients are not periodic, then $N(\Lambda)>0$, but there is no diagonal operator $D$ such that $D\Lambda$ is an algebraic lattice. Because of Dirichlet’s theorem on algebraic units, algebraic lattices have rich symmetry groups consisting of diagonal operators. Details can be found, for instance, in [83] and [19], and also in [90]. This observation generalises the fact that quadratic irrationalities have periodic continued fractions to the multidimensional case. Thus, Cassels–Swinnerton-Dyer’s conjecture claims that for $d\geqslant3$ the inequality $N(\Lambda)>0$ (which, we recall, is an analogue of the property of a number to be badly approximable) implies that $\Lambda$ has a rich group of symmetries consisting of diagonal operators. This statement admits a reformulation in terms of multidimensional continued fractions — to be more specific, in terms of Klein polyhedra. Definition 28. Let $\Lambda$ be a full-rank lattice in $\mathbb{R}^d$. In each orthant $\mathcal{O}$ consider the convex hull Each of the $2^d$ convex hulls thus obtained is called a Klein polyhedron. In the case $d=2$ Definition 28 differs from Definition 6 of Klein polygons given in § 2.5.4. The reason is that Klein polygons from Definition 6 are a geometric interpretation of a continued fraction of a single number. If we take a pair of distinct numbers, then the corresponding pair of lines divides the plane into four angles and the corresponding pair of linear forms determines a full-rank lattice in $\mathbb{R}^2$. This construction precisely corresponds to Definition 28, but it interprets geometrically the continued fractions of a pair of numbers. We discuss it in detail in § 7.3. We say that a lattice $\Lambda$ is irrational if none of its non-zero points has zero coordinates. If a lattice is irrational, then each of the $2^d$ Klein polyhedra is a generalised polyhedron, that is, a set whose intersection with any given compact polyhedron is itself a compact polyhedron. This was proved in [91]. In particular, in this case each vertex belongs to a finite number of faces. If, at the same time, the dual lattice $\Lambda^\ast$ is irrational, then, as it shown in [92], each of the $2^d$ Klein polyhedra has only compact faces. In particular, each face has a finite number of vertices. We showed in § 2.5.4 that in the two-dimensional case the role of partial quotients is played by edges and vertices of Klein polygons equipped with linear integer invariants such as the integer length of an edge and the integer angle at a vertex. In the multidimensional case it is natural to consider facets (that is, $(d-1)$-dimensional faces) and edge stars of vertices of a Klein polyhedron equipped with some appropriate linear integer invariants. In [92]–[95] the determinants of facets and edge stars were considered as such invariants. Definition 29. Let $\Lambda$ be a full-rank lattice in $\mathbb{R}^d$. Let $\Lambda$ and $\Lambda^\ast$ be irrational. Let $\mathcal{K}$ be one of the $2^d$ Klein polyhedra of $\Lambda$. (i) Let $F$ be an arbitrary facet of $\mathcal{K}$, and let $\mathbf{v}_1,\dots,\mathbf{v}_k$ be the vertices of $F$. Then the determinant of $F$ is defined by
$$
\begin{equation*}
\det F=\sum_{1\leqslant i_1<\cdots<i_d\leqslant k} |\!\det(\mathbf v_{i_1},\dots,\mathbf v_{i_d})|.
\end{equation*}
\notag
$$
(ii) Let $\mathbf{v}$ be an arbitrary vertex of $\mathcal{K}$ and let it be incident to $k$ edges. Let $\mathbf{r}_1,\dots,\mathbf{r}_k$ be primitive vectors of $\Lambda$ parallel to these edges. Then the determinant of the edge star $\operatorname{St}_{\mathbf v}$ of $\mathbf{v}$ is defined by It is clear that if $d=2$ and $\det\Lambda=1$, then the determinants of edges are equal to their integer lengths and the determinants of the edge stars of vertices are equal to the integer angles between the corresponding edges. According to Corollary 6 in § 2.5.1, the property of an irrational number to be badly approximable is equivalent to its property to have bounded partial quotients. The following multidimensional generalisation of this statement was proved in [92]–[94]. Theorem 39 (German, 2006). Let $\Lambda$ be an irrational full-rank lattice in $\mathbb{R}^d$. Then the following statements are equivalent: (i) $N(\Lambda)>0$; (ii) the facets of all the $2^n$ Klein polyhedra of $\Lambda$ have bounded (by a common constant) determinants; (iii) the facets and the edge stars of the vertices of the Klein polyhedron corresponding to $\Lambda$ and the positive orthant have bounded (by a common constant) determinants. The corollary to Dirichlet’s theorem on algebraic units mentioned above can be elaborated further. If $\Lambda$ is an algebraic lattice in $\mathbb{R}^d$, then there is a group isomorphic to $\mathbb{Z}^{d-1}$ which consists of diagonal operators with positive diagonal elements that preserve the lattice. In this case the boundary of each of the $2^d$ Klein polyhedra equipped with any linear integer invariants that can be of interest to us has a $(d-1)$-periodic combinatorial structure. Thus, Theorem 39 provides the following reformulation of Cassels–Swinnerton- Dyer’s conjecture. Cassels–Swinnerton-Dyer’s Conjecture (reformulated). Let $d\geqslant3$ and let $\Lambda$ be an irrational full-rank lattice in $\mathbb{R}^d$. Let $\mathcal{K}$ be one of the $2^n$ Klein polyhedra of $\Lambda$. Then the following statements are equivalent: (i) the facets and the edge stars of the vertices of $\mathcal{K}$ have bounded (by a common constant) determinants; (ii) the combinatorial structure of the boundary of $\mathcal{K}$ equipped with determinants of facets and determinants of edge stars of vertices is $(d-1)$-periodic. Thus, we can say that Cassels–Swinnerton-Dyer’s conjecture claims that for $d\geqslant 3$ the boundedness of the multidimensional analogues of partial quotients implies the periodicity of the corresponding multidimensional continued fraction. 7.3. A relationship with the growth of multidimensional analogues of partial quotientsIf statement (ii) or statement (iii) of Theorem 39 does not hold, then $N(\Lambda)=0$, that is, $\Pi(\mathbf{z})$ attains arbitrarily small values at non-zero $\mathbf{z}\in\Lambda$. In the two-dimensional case we can recall Corollary 7 in § 2.5.1, which relates the Diophantine exponent of a number to the growth of its partial quotients. The correspondence described in § 2.5.2 enables one to formulate a statement equivalent to Corollary 7 in terms of Diophantine exponents of lattices in dimension $2$. Let us extend Definition 6 of Klein polygons to the case of two numbers $\theta_1$ and $\theta_2$ by considering the convex hulls of the non-zero integer points in the four angles defined by the lines $y=\theta_1x$ and $y=\theta_2x$ (see Fig. 11). In the case of a single number, a partial quotient $a_{k+1}$ equals the integer angle at the vertex $\mathbf{v}_k$ of the corresponding Klein polygon, whereas the denominator $q_k$ differs from $|\mathbf{v}_k|$ by a bounded factor. Hence, in the case of two numbers, relation (4) in Corollary 7 provides the equality
$$
\begin{equation}
\max\bigl(\omega(\theta_1),\omega(\theta_2)\bigr)= 1+\limsup_{\substack{\mathbf w\in \mathcal{W},\ |\mathbf w|>1 \\ |\mathbf w|\to\infty}}\frac{\log(\det\operatorname{St}_{\mathbf w})}{\log|\mathbf w|}\,,
\end{equation}
\tag{146}
$$
where $\mathcal{W}$ denotes the set of vertices of all four extended Klein polygons. Now let us address the Diophantine exponent of the lattice $\Lambda=A\mathbb{Z}^2$, where
$$
\begin{equation*}
A=\begin{pmatrix} \theta_1 & -1 \\ \theta_2 & -1 \end{pmatrix}.
\end{equation*}
\notag
$$
Denote by $\mathcal{V}$ the set of vertices of all four Klein polyhedra of $\Lambda$. Since $\mathcal{V}=A\mathcal{W}$, we have the following relations for $\mathbf{v}=A(\mathbf{w})$:
$$
\begin{equation}
|\mathbf v|\asymp|\mathbf w| \quad\text{and}\quad \det\operatorname{St}_{\mathbf w}\asymp\det\operatorname{St}_{\mathbf v}.
\end{equation}
\tag{147}
$$
If $\mathbf{w}=(q,p)$, then for $|q|$ large enough
$$
\begin{equation}
\Pi(\mathbf v)\asymp|\mathbf v|^{-\gamma} \ \ \iff \ \ \min\bigl(|q\theta_1-p|,|q\theta_2-p|\bigr)\asymp q^{-1-2\gamma}.
\end{equation}
\tag{148}
$$
Further, if all the vertices of all the Klein polyhedra of $\Lambda$ satisfy the condition $\Pi(\mathbf{v})\geqslant|\mathbf{v}|^{-\gamma}$, then so do all the non-zero points of $\Lambda$. This takes us to the following equality:
$$
\begin{equation*}
\omega(\Lambda)=\sup\{\gamma\in\mathbb{R}\mid \text{ there exist infinitely many }\mathbf v \in\mathcal{V}\text{ such that }\Pi(\mathbf v)\leqslant|\mathbf v|^{-\gamma}\}.
\end{equation*}
\notag
$$
Hence by (148)
$$
\begin{equation*}
\max\bigl(\omega(\theta_1),\omega(\theta_2)\bigr)=1+2\omega(\Lambda).
\end{equation*}
\notag
$$
Taking (146) and (147) into account we obtain
$$
\begin{equation}
\omega(\Lambda)= \frac{1}{2}\limsup_{\substack{\mathbf v\in \mathcal{V},\ |\mathbf v|>1 \\ |\mathbf v|\to\infty}}\frac{\log(\det\operatorname{St}_{\mathbf v})} {\log|\mathbf v|}\,.
\end{equation}
\tag{149}
$$
Thus, relation (149) is an analogue of equality (4) in Corollary 7 for Diophantine exponents of lattices of rank $2$. It can be generalised to the multidimensional case, but what is currently known is far from an exhaustive generalisation. There is only one result, obtained in [95]. Theorem 40 (Bigushev and German, 2022). Let $\Lambda$ be an irrational full-rank lattice in $\mathbb{R}^3$. Let $\mathcal{V}$ denote the set of vertices of all the eight Klein polyhedra of $\Lambda$. Then It was shown in the same paper [95] that the main statement upon which the proof of Theorem 40 is based cannot be inverted because there is a counterexample. Nevertheless, this statement is perhaps too local and its locality can be ‘relaxed’ slightly. Another way that can probably help to turn inequality (150) to equality is to find a better quantitative characteristic of multidimensional partial quotients than determinants. 7.4. A transference theoremIn the case of Diophantine exponents of lattices, there also is a transference theorem. Just as with the theorems of Khintchine, Dyson, and their generalisations for multiplicative exponents and Diophantine approximation with weights, the simplest way to prove it is to derive it from Mahler’s theorem in disguise of Theorem 22. As before, we denote by $\Lambda^\ast$ the dual lattice of $\Lambda$. For $d=2$ the lattice $\Lambda^\ast$ coincides with $\Lambda$ rotated by $\pi/2$ up to homothety. Thus, in the two-dimensional case we have the obvious equality $\omega(\Lambda)=\omega(\Lambda^\ast)$. The following theorem was proved in [87]. Theorem 41. If one of the exponents $\omega(\Lambda)$ and $\omega(\Lambda^\ast)$ vanishes, then so does the other. If both are non-zero, then Let us show how to derive Theorem 41 from Theorem 22. For each tuple $\boldsymbol{\lambda} =(\lambda_1,\dots,\lambda_d)\in\mathbb{R}_+^d$, let us define the parallelepiped $\mathcal{P}(\boldsymbol{\lambda})$ by
$$
\begin{equation}
\mathcal{P}(\boldsymbol{\lambda})=\bigl\{\mathbf z=(z_1,\dots,z_d)\in\mathbb{R}^d\bigm| |z_i|\leqslant\lambda_i,\ i=1,\dots,d\bigr\}.
\end{equation}
\tag{152}
$$
Let us assign to each tuple $\boldsymbol{\lambda}$ the tuple
$$
\begin{equation}
\boldsymbol{\lambda}^\ast=(\lambda_1^\ast,\dots,\lambda_d^\ast),\quad\text{where } \lambda_i^\ast=\lambda_i^{-1}\Pi(\boldsymbol{\lambda})^d, \quad i=1,\dots,d.
\end{equation}
\tag{153}
$$
Then $\mathcal{P}(\boldsymbol{\lambda})^\ast =\mathcal{P}(\boldsymbol{\lambda}^\ast)$, that is, $\mathcal{P}(\boldsymbol{\lambda}^\ast)$ is the pseudocompound of $\mathcal{P}(\boldsymbol{\lambda})$ (see Definition 14 in § 3.4.2). By Theorem 22
$$
\begin{equation}
\mathcal{P}(\boldsymbol{\lambda}^\ast)\cap\Lambda^\ast\ne \{\mathbf 0\}\ \ \implies\ \ (d-1)\mathcal{P}(\boldsymbol{\lambda})\cap\Lambda\ne \{\mathbf 0\}.
\end{equation}
\tag{154}
$$
Next, for each $\delta\geqslant0$, set If $\Pi(\boldsymbol{\lambda}^\ast)=|\boldsymbol{\lambda}^\ast|^{-\delta}$, then
$$
\begin{equation*}
\Pi(\boldsymbol{\lambda})=\Pi(\boldsymbol{\lambda}^\ast)^{1/(d-1)}= |\boldsymbol{\lambda}^\ast|^{-\delta/(d-1)}
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
|\boldsymbol{\lambda}|\leqslant\frac{\Pi(\boldsymbol{\lambda})^d} {\min_{1\leqslant i\leqslant d}|\lambda_i|^{d-1}}= \frac{|\boldsymbol{\lambda}^\ast|^{d-1}}{\Pi(\boldsymbol{\lambda})^{d(d-2)}}= |\boldsymbol{\lambda}^\ast|^{d-1+d(d-2)\delta/(d-1)}.
\end{equation*}
\notag
$$
Therefore,
$$
\begin{equation}
\Pi(\boldsymbol{\lambda}^\ast)=|\boldsymbol{\lambda}^\ast|^{-\delta}\ \ \implies\ \ \Pi(\boldsymbol{\lambda})\leqslant|\boldsymbol{\lambda}|^{-\gamma}.
\end{equation}
\tag{155}
$$
By (155) and (154) we have
$$
\begin{equation*}
\omega(\Lambda^\ast)\geqslant\delta\ \ \implies\ \ \omega(\Lambda)\geqslant\gamma=\frac{\delta}{(d-1)^2+d(d-2)\delta}\,,
\end{equation*}
\notag
$$
so that
$$
\begin{equation*}
\omega(\Lambda)\geqslant \frac{\omega(\Lambda^\ast)}{(d-1)^2+d(d-2)\omega(\Lambda^\ast)}
\end{equation*}
\notag
$$
or, equivalently,
$$
\begin{equation*}
\frac{1+\omega(\Lambda)^{-1}}{1+\omega(\Lambda^\ast)^{-1}}\leqslant (d-1)^2.
\end{equation*}
\notag
$$
Since $\Lambda$ and $\Lambda^\ast$ can be swapped, we obtain both inequalities in (151). 7.5. Parametric geometry of numbersIn § 3.4.8 we described an approach developed by Schmidt and Summerer. A fundamental role in this approach is played by the space of parameters
$$
\begin{equation*}
\mathcal{T}=\bigl\{\boldsymbol{\tau}=(\tau_1,\dots,\tau_d)\in \mathbb{R}^d\bigm| \tau_1+\cdots+\tau_d=0 \bigr\}.
\end{equation*}
\notag
$$
We saw in §§ 3.4.8 and 4.4.4 that the problem of approximating zero with the values of several linear forms requires considering one-dimensional subspaces of $\mathcal{T}$ along which $\boldsymbol{\tau}$ is to tend to infinity. As for Diophantine approximation with non-trivial weights, we saw in § 6.4 that it requires working with two-dimensional subspaces of $\mathcal{T}$. In the context of problems related to Diophantine exponents of lattices, we need to work with the whole of $\mathcal{T}$. For each $\boldsymbol{\tau}\in\mathcal{T}$ let us set
$$
\begin{equation*}
|\boldsymbol{\tau}|_+=\max_{1\leqslant i\leqslant d}\tau_i.
\end{equation*}
\notag
$$
Every norm in $\mathbb{R}^d$, for instance, the sup-norm $|\,{\cdot}\,|$, induces a norm in $\mathcal{T}$. But the functional $|\,{\cdot}\,|_+$ is not a norm for $d\geqslant3$, as the corresponding ‘unit balls’ are simplices, which are obviously not symmetric about the origin. Nevertheless, the functional $|\,{\cdot}\,|_+$ plays a very important role, as it is the image of the sup-norm under the logarithmic mapping: if
$$
\begin{equation*}
\mathbf z=(z_1,\dots,z_d),\qquad z_i>0,\quad i=1,\dots,d,
\end{equation*}
\notag
$$
and then It is clear that $|\,{\cdot}\,|_+$ generates a monotone exhaustion of $\mathcal{T}$, that is, $\mathcal{T}=\bigcup_{\lambda>0}\mathcal{S}(\lambda)$, where
$$
\begin{equation*}
\mathcal{S}(\lambda)=\{\boldsymbol{\tau}\in\mathcal{T}\mid f(\boldsymbol{\tau})\leqslant \lambda\}\quad\text{for }\lambda>0,
\end{equation*}
\notag
$$
each set $\mathcal{S}(\lambda)$ is compact, and $\mathcal{S}(\lambda')$ is contained in the (relative) interior of $\mathcal{S}(\lambda'')$ for $\lambda'<\lambda''$. In particular, $|\boldsymbol{\tau}|_+\to+\infty$ is equivalent to $|\boldsymbol{\tau}|\to+\infty$. Diophantine exponents of lattices require the following modification of Definition 17. Definition 30. Given a lattice $\Lambda$ and $k\in\{1,\dots,d\}$, the quantities Definition 31. Given a lattice $\Lambda$ and $k\in\{1,\dots,d\}$, the quantities It was shown in [68] that the Diophantine exponent of $\Lambda$ is related to $\underline{\varphi}_1(\Lambda)$ by
$$
\begin{equation*}
\bigl(1+\omega(\Lambda)\bigr)\bigl(1+\underline{\varphi}_1(\Lambda)\bigr)=1.
\end{equation*}
\notag
$$
This relation is equivalent to
$$
\begin{equation*}
1+\omega(\Lambda)^{-1}=-\underline{\varphi}_1(\Lambda)^{-1}.
\end{equation*}
\notag
$$
Thus, inequalities (151) can be rewritten as follows:
$$
\begin{equation*}
(d-1)^{-2}\leqslant \frac{\underline{\varphi}_1(\Lambda^\ast)}{\underline{\varphi}_1(\Lambda)} \leqslant (d-1)^2.
\end{equation*}
\notag
$$
And just as with Khintchine’s and Dyson’s inequalities, as well as with transference inequalities for Diophantine approximation with weights, this approach enables one to split inequality (151) into a chain of inequalities between intermediate exponents. The corresponding definitions and formulations can be found in [68]. 7.6. The uniform exponent and Mordell’s constantThe attentive reader may have noticed that in all previous sections we discussed two types of Diophantine exponents, regular and uniform ones. But in the case of Diophantine exponents of lattices we have discussed only regular exponents so far, without mentioning their uniform analogues at all. There is a reason for this, and we discuss this reason in the current subsection. The thing is that, just as with the problem of approximating a real number by rationals (see § 2.6), the uniform exponent of a lattice is ‘trivial’: it is either infinity, or zero. As a uniform analogue of $\omega(\Lambda)$ it is natural to consider the quantity $\widehat\omega(\Lambda)$ equal to the supremum of the real $\gamma$ such that for $t$ large enough and every $\boldsymbol{\lambda}\in\mathbb{R}_+^d$ satisfying the relations
$$
\begin{equation*}
|\boldsymbol{\lambda}|=t \quad\text{and}\quad \Pi(\boldsymbol{\lambda})=t^{-\gamma}
\end{equation*}
\notag
$$
the parallelepiped $\mathcal{P}(\boldsymbol{\lambda})$ defined by (152) contains a non-zero point of $\Lambda$. 7.6.1. Mordell’s constantIn 1937 Mordell [96] asked whether or not there exists a constant $c$ depending only on dimension $d$ such that for every lattice $\Lambda$ with determinant $1$ there is a tuple $\boldsymbol{\lambda}=(\lambda_1,\dots,\lambda_d)\in\mathbb{R}_+^d$ satisfying the following two conditions:
$$
\begin{equation*}
\prod_{i=1}^{d}\lambda_i=c\quad\text{and}\quad \mathcal{P}(\boldsymbol{\lambda})\cap\Lambda=\{\mathbf 0\},
\end{equation*}
\notag
$$
where $\mathcal{P}(\boldsymbol{\lambda})$ is the parallelepiped defined by (152). In the same year Siegel showed in a letter to Mordell that the answer to this question is positive. A slightly different proof was proposed by Davenport in [97]. Davenport proved the following statement. Theorem 42 (Davenport, 1937). Let $\Lambda$ be a full-rank lattice in $\mathbb{R}^d$ with determinant $1$. Consider an arbitrary $d$-tuple $\boldsymbol{\lambda} =(\lambda_1,\dots,\lambda_d)\in\mathbb{R}_+^d$ such that Let $\mu_k=\mu_k\bigl(\mathcal{P}(\boldsymbol{\lambda}),\Lambda\bigr)$, $k=1,\dots,d$, denote the $k$th successive minimum of the parallelepiped $\mathcal{P}(\boldsymbol{\lambda})$ with respect to $\Lambda$. Then there is a positive constant $c$ depending only on $d$ and a permutation $k_1,\dots,k_d$ of the indices $1,\dots,d$ such that the interior of $\mathcal{P}(\boldsymbol{\lambda}')$, where contains no non-zero points of $\Lambda$. It follows from Minkowski’s second theorem (see Theorem 24 in § 3.4.8) that the tuple $\boldsymbol{\lambda}'$ in Theorem 42 satisfies Thus, Theorem 42, gives indeed a positive answer to Mordell’s question. Correspondingly, if for each lattice $\Lambda$ in $\mathbb{R}^d$ with determinant $1$ we define its Mordell’s constant by
$$
\begin{equation*}
\kappa(\Lambda)=\sup_{\substack{\boldsymbol{\lambda}=(\lambda_1,\dots,\lambda_d)\in \mathbb{R}_+^d \\ \mathcal{P}(\boldsymbol{\lambda})\cap\Lambda=\{\mathbf 0\}}}\, \prod_{i=1}^{d}\lambda_i,
\end{equation*}
\notag
$$
then the quantity
$$
\begin{equation*}
\kappa_d=\inf_{\Lambda\colon\det\Lambda=1}\kappa(\Lambda)
\end{equation*}
\notag
$$
is positive for each $d$. The best known estimate belongs to Shapira and Weiss [98]. Their result confirms the conjecture, made by Ramharter [99], that $\limsup_{d\to\infty}\kappa_d^{1/d\log d}>0$. 7.6.2. The triviality of the uniform exponentTheorem 42 has a local nature. It enables one, therefore, to find ‘empty’ parallelepipeds of fixed volume arbitrarily ‘far away’. Let us assume for simplicity that the first coordinate axis
$$
\begin{equation*}
\bigl\{\mathbf z=(z_1,\dots,z_d)\in\mathbb{R}^d \bigm| z_i=0,\ i=2,\dots,d\bigr\}
\end{equation*}
\notag
$$
is not a subset of any subspace $\mathcal{L}$ of dimension $k<d$ such that $\mathcal{L}\cap\Lambda$ is a lattice of rank $k$. Then for every $\varepsilon>0$ the cylinder
$$
\begin{equation*}
\bigl\{\mathbf z=(z_1,\dots,z_d)\in\mathbb{R}^d \bigm| |z_i|<\varepsilon,\ i=2,\dots,d\bigr\}
\end{equation*}
\notag
$$
contains $d$ linearly independent points of $\Lambda$. Thus, for every such $\varepsilon$ there is $\lambda\in\mathbb{R}_+$ and a tuple $\boldsymbol{\lambda}=(\lambda_1,\dots,\lambda_d)\in\mathbb{R}_+^d$ such that
$$
\begin{equation*}
\lambda_1=\lambda, \qquad \lambda_i=\lambda^{-1/(d-1)}, \quad i=2,\dots,d,
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\mu_d\bigl(\mathcal{P}(\boldsymbol{\lambda}),\Lambda\bigr)\lambda^{-1/(d-1)} <\varepsilon.
\end{equation*}
\notag
$$
By Theorem 42 the parallelepiped $\mu_d\bigl(\mathcal{P}(\boldsymbol{\lambda}), \Lambda\bigr)\mathcal{P}(\boldsymbol{\lambda})$ contains a parallelepiped $\mathcal{P}(\boldsymbol{\lambda}')$ of fixed volume that does not contain non-zero points of $\Lambda$. It immediately follows that $\widehat\omega(\Lambda)=0$.
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\Bibitem{Ger23} |
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