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Uniqueness sets of positive measure for the trigonometric system M. G. Plotnikovab a Lomonosov Moscow State University, Faculty of Mechanics and Mathematics b Moscow Center for Fundamental and Applied Mathematics
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     § 1. IntroductionIn the paper, the problems are studied on uniqueness of functions expansion to an orthogonal series. One of the remarkable topis in this area is the theory of uniqueness sets, originating from Cantor’s studies on the nature of sets outside of which the uniqueness is not violated when trigonometric series converge. This research led Cantor to the creation of set theory. In the book [1], which contains the set-theoretical works of G. Cantor with notes by E. Zermelo, Zermelo writes that “in the theory of trigonometric series, we see the birthplace of Cantor’s set theory”. Later, deep relationships were revealed between the theory of uniqueness sets and other branches of mathematics: descriptive set theory, functional analysis, abstract harmonic analysis and group representation theory, number theory. The development of this theory was stimulated by a large number of difficult problems, that remained and remain open for many decades (see [2], [3], [4], Ch. 11, [5], Ch. 14). A set $A \subset D$ is called a uniqueness set or $U$-set for some orthogonal system of functions $\{f_n\}$ having a common domain $D$, if the convergence of the series $\sum_n c_n f_n$ to zero on $D \setminus A$ implies that all $c_n =0$. If every series $\sum_n c_n f_n$ converging on $D \setminus A$ to a finite integrable function needs to be its Fourier series, then the set $A$ is called a $V$-set (sometimes, $U^*$-set) for the system $\{f_n\}$. It is obvious, that every $V$-set is a $U$-set. Whether the converse statement is true is an open question. The widest known class of $V$-sets for the trigonometric system is found in [6]. If $A \subset D$ is not a $U$-set, it is called a $M$-set. Denote by $\mathcal{U}$ and $\mathcal{M}$ the classes of such sets for the trigonometric system $\{\exp(i n x)\}_{n \in \mathbb{Z}}$ with the domain $\mathbb{T}=[-\pi, \pi)$. Among sets of zero measure there are sets both from $\mathcal{U}$ and from $\mathcal{M}$. The usual methods of classifying sets of measure zero according to their “density”, such as capacities or Hausdorff dimensions, do not distinguish the classes $\mathcal{U}$ and $\mathcal{M}$. The answer to the question, $A \in \mathcal{U}$ or $A \in \mathcal{M}$, is connected both with the metric and topological, and algebraic properties of the set $A$ inside $\mathbb{T}$ as a group. Even in the simplest cases, algebraic number theory is used to answer this question. Salem–Zygmund(–Pyatetsky-Shapiro–Bari) theorem states that if $F_\zeta$ is the symmetric perfect set with a constant ratio $\zeta$, then $F$ is a $U$-set if and only if $\zeta^{-1}$ is a Piso number (see [2], [4], Ch. 11, [5], Ch. 14). On the other hand, a complete characterization of $U$-sets cannot be obtained in a constructive way [2]. If $A$ is a measurable set having positive measure, then it is easy to show that $A \in \mathcal{M}$, and hence $A$ is not a $V$-set. This means that it is impossible to unambiguously recover the complete appearance of the trigonometric series, having information about its convergence only on a set of incomplete measure and, a fortiori on a set of arbitrarily small measure. Still, it is interesting to know, at what settings this can be done. Our work belongs to this direction. There are few known results here; usually only $U$-sets are considered, and only under some restrictions. One of these results is Zygmund’s theorem on sets of relative uniqueness ([4], Ch. 11): if $b_n \downarrow 0$, then for the (Zygmund) class of trigonometric series $\sum_{n \in \mathbb{Z}} c_n \exp(i n x)$ with $|c_{|n|} | \leqslant b_n$ and for each $\delta>0$ there exist $U$-sets of measure greater than $2 \pi-\delta$. The above theorem was strengthened in different directions in [7], [8]. For each Zygmund class $\Lambda$ in [7] a class $U(\Lambda)$-sets of full measure is constructed, and in [8] a class $V(\Lambda)$-sets of positive measure (we call the latter $\mathcal{V}(\Lambda)$). It was also shown in [8] that each function $f \in L(\mathbb{T})$ with the Fourier series from $\Lambda$ can be uniquely recovered from its values on a set of the form $\mathbb{T} \setminus V$, $V \in \mathcal{V}(\Lambda)$. The last result is not about recovering a series from a function, but about recovering a function from its part! $U$-sets of positive measure can also be obtained by considering some subsystem $\{f_{n_k}\}$ instead of the system of functions $\{f_n\}$ as a whole. This approach is close to Zygmund one in some sense, since the set of series $\sum c_{n_k} f_{n_k}$ is a subset of the set of series $\sum c_n f_n$. In the nontrigonometric case, the Rademacher system and the system of Rademacher $d$-chaos, which are natural subsystems of the Walsh system ($=: \{f_n\}$), are often considered suitable candidates for the role of $\{f_{n_k}\}$. $U$-sets of positive measure for such $\{f_{n_k}\}$ are found in [9]–[12]; at the same time, there are no such sets for the entire Walsh system. We are not aware on results of this kind for subsystems of the trigonometric system. In [13] another approach, not related to restrictions on the class of series, was applied to the problem of recovering a series from its sum taken on a set of small measure. In [13], for the first time, complete orthonormal systems with $U$-sets of positive measure were indicated. They turned out to be the special rearrangements of the Vilenkin–Chrestenson systems. The main goal of this paper is to find the class $\mathcal{B}$ of rearrangements of the trigonometric system $\{\exp(i n x)\}_{n \in \mathbb{Z}}$ for which there are $U$-sets of positive measure. (Further on, we identify rearrangements of the trigonometric system with rearrangements of the set $\mathbb{Z}$.) The goal is reached in § 5 (see Theorem 10). Along the way, the class $\mathcal{B}_{\mathcal{M}} \supset \mathcal{B}$ is found, for which it is proved (Theorem 3) that $\varnothing$ is a $V$-set for the system $\{\exp(i B(n) x)\}_{n \in \mathbb{Z}}$ if $B \in \mathcal{B}_{\mathcal{M}}$ (for the connection between this theorem and the Stechkin–Ulyanov conjecture, see § 6). Let us present some other results of the paper. In Theorem 8 the following effect is discovered. There exists a countable subgroup $H \subset \mathbb{T}$ such that if one means by convergence of a trigonometric series its usual convergence on $\mathbb{T} \setminus H$, and convergence of its $B$-rearrangements, $B \in \mathcal{B}$, on $H$, then, with such a small “reconfiguration”, $U$-sets of positive measure are already appeared. Theorem 6 strengthens the main theorem 10 as follows: if $B \in \mathcal{B}$ and the series $BTS$ in the system $\{\exp(i B(n) x)\}_{n \in \mathbb{Z}}$ converges on some special set $A$ of arbitrarily small measure to a finite function $f \in L(\mathbb{T})$, then its coefficients can be compute, knowing only the values of $f$ on $A$ (see (5.19)). Let us give a brief survey of the most interesting, in our opinion, recent papers on the theory of uniqueness. Kozma and Olevsky constructed [14] an example of a non-trivial trigonometric series with coefficients tending to zero, which converges to zero everywhere in a subsequence of partial sums. Thus, the long-standing problem of Ul’yanov was solved [15]. Gevorkyan constructed [16] new classes of $U$-sets for the multidimensional trigonometric systems. The categorical properties of $U$-sets were studied by Kholshchevnikova and Skvortsov [17], [18]. The papers [19]–[25] are devoted to the study of uniqueness problems for the Walsh and Franklin systems as well as the systems of characters of zero-dimensional groups. Our work is organized as follows. In § 2 the main definitions and notation are given, and a number of auxiliary results are proved. In § 3 the notion of $D$-monotonicity of permutations of the set $\mathbb{Z}$ was introduced and studied, which characterizes the fact that the mapping $B \colon \mathbb{Z} \to \mathbb{Z }$ “not too often” changes the direction of monotonicity. In § 4, candidates for the roles of the class $\mathcal{B}$ as well as of $U$-sets of positive measure corresponding to this class are selected. The main results of the work are presented in § 5. Finally, § 6 contains a small discussion of some results of the work and open problems. § 2. Definitions, notation, and auxiliary results2.1. Main definitions and notationsBy $\# A$ we denote the cardinality of a finite set $A$. Let $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{R}$, and $\mathbb{C}$ be the set of positive integers, non-negative integers, integers, positive reals, and complex numbers, respectively,
$$
\begin{equation*}
\operatorname{sinc}(x):= \begin{cases} \dfrac{\sin(x)}{x} &\text{if }x \ne 0, \\ 1 &\text{if }x=0. \end{cases}
\end{equation*}
\notag
$$
The symbol $\mathbb{T}$ means in this work both simply the half-open interval $[-\pi, \pi)$ and the one-dimensional torus, i. e. the (topological) abelian group consisting of the set $[-\pi, \pi)$ and the group operation $\oplus$: $x \,{\oplus}\, y := \operatorname{arg} \exp(i(x+y))$. Here $\operatorname{arg} z$ is the argument of the complex number $z \ne 0$, $\operatorname{arg} z \in [-\pi, \pi)$. Suppose we are given an interval $I =(a,b)$, where $- \pi<a<b<\pi$, and a function $G \colon \mathbb{T} \to \mathbb{C}$. Set
$$
\begin{equation*}
\Delta^2 G(I):= G(b)-2 G\biggl(\frac{a+b}{2}\biggr)+G(a).
\end{equation*}
\notag
$$
It is known that the trigonometric system of functions $\{\exp(i n x)\}_{n \in \mathbb{Z}}$ is complete and orthogonal in $L_2(\mathbb{T})$. We study series
$$
\begin{equation}
TS=\sum_{n\in\mathbb{Z}} c_n \exp(inx),\qquad c_n \in \mathbb{C},
\end{equation}
\tag{2.1}
$$
with respect to this system, as well as their rearrangements. If a bijection $B \colon \mathbb{Z} \to \mathbb{Z}$ is given, then by the $B$-rearranged series $TS$ we mean the series The rule $TS \mapsto BTS$ establishes a one-to-one correspondence between the sets of series with respect to the systems $\{\exp(i n x)\}_{n \in \mathbb{Z}}$ and $\{\exp(i B(n) x)\}_{n \in \mathbb{Z}}$. If $N \in \mathbb{N}$, then the sum $S_N=\sum_{n=-N}^N$ of the corresponding terms of a series (2.1) or (2.2) is called its $N$ th partial sum. The convergence of the series to a value $A \in \mathbb{C}$ at a point $x \in \mathbb{T}$ means that $\lim_{N \to \infty} S_N(x)=A$. We say that the bijection $B \colon \mathbb{Z} \to \mathbb{Z}$ is odd if $B(- n)=- B(n)$ for every $n \in \mathbb{N} \cup \{0\}$, in particular, $B(0)=0$. If the bijection $B$ is odd, then each series (2.2) can be written using sines and cosines as follows:
$$
\begin{equation}
\frac{a_0}{2}+\sum_{n=1}^\infty\bigl(a_{B(n)} \cos(B(n) x)+b_{B(n)} \sin(B(n) x)\bigr).
\end{equation}
\tag{2.3}
$$
The coefficients of the series (2.3) are found as follows:
$$
\begin{equation}
a_{B(n)}=c_{B(n)}+c_{B(- n)},\qquad b_{B(n)}=i(c_{B(n)}-c_{B(- n)}), \qquad a_0=2 c_0.
\end{equation}
\tag{2.4}
$$
It is easy to see that formula (2.4) establishes a linear isomorphism between the sets of series (2.2) and (2.3). Moreover, in this case
$$
\begin{equation}
\sum_{n=-N}^N c_{B(n)} \exp(i B(n) x)\equiv \frac{a_0}{2}+\sum_{n=1}^N \bigl( a_{B(n)} \cos(B(n) x)+b_{B(n)} \sin(B(n) x)\bigr)
\end{equation}
\tag{2.5}
$$
for all $N \in \mathbb{N}$. It yields from (2.5) that if the bijection $B$ is odd, then the convergence of the series (2.2) is equivalent to the convergence of the corresponding series (2.3) to the same sum. Since the main results of the paper are formulated for $B$-rearranged trigonometric series in the exponential form and odd bijections of $B$, they can be easily extended to the case of the cosine-sine form. 2.2. Trigonometric series and their Riemann functionsHere we consider trigonometric series (2.1) with coefficients tending to zero. Each such series generates its own Riemann function
$$
\begin{equation}
F(x):= \frac{c_0 x^2}{2}-\sum_{n \in \mathbb{Z} \setminus \{0\}}\frac{c_n}{n^2} \exp(inx),
\end{equation}
\tag{2.6}
$$
which is obtained as a result of its double formal integration. The series on the right-hand side of (2.6) converges absolutely and uniformly in $x$, and the function $F$ is continuous on $\mathbb{T}$. It is known that
$$
\begin{equation}
\frac{F(x+2h)-2 F(x)+F(x-2h)}{4 h^2}=\sum_{n\in\mathbb{Z}} c_n \exp(inx) \operatorname{sinc}^2(nh)
\end{equation}
\tag{2.7}
$$
(see, for example, the proof of Theorem 1.1 in [26]). The second symmetric derivative (the second Schwartz derivative) of a function $G$ at a point $x \in(-\pi, \pi)$ is defined by
$$
\begin{equation}
D^2_{\mathrm{symm}} G(x)= \lim_{h \to 0}\frac{G(x+h)-2G(x)+G(x-h)}{h^2}.
\end{equation}
\tag{2.8}
$$
If the usual second derivative $D^2 G(x)$ exists, then $D^2_{\mathrm{symm}} G(x)$ also exists and $D^2_{\mathrm{symm}} G(x)=D^2 G(x)$. The following de la Vallee-Poussin theorem deals with the possibility of recovering the function $G(x)$ from $D^2_{\mathrm{symm}} G(x)$ (see, for example, the proof of Lemma (iv) in [4], Ch. 11, 11.31. Theorem A1. Let a function $f$ be finite and summable on $(a,b)$, and a function $F$ be continuous on the interval $[a,b]$, and $D^2_{\mathrm{symm}} F(x)=f(x)$ for all $x \in(a,b)$. Then there exist constants $A$ and $B$ such that A series (2.1) is called $R$-summable to a value $A$ at a point $x_0$ if
$$
\begin{equation}
\lim_{h \to 0}\biggl( \sum_{n\in\mathbb{Z}} c_n \exp(i n x_0) \operatorname{sinc}^2(n h) \biggr)=A.
\end{equation}
\tag{2.9}
$$
Relation (2.9) is equivalent to
$$
\begin{equation}
\lim_{h \to 0} \biggl( \sum_{n\in\mathbb{Z}} c_{B(n)} \exp(iB(n)x_0) \operatorname{sinc}^2(B(n)h) \biggr) =A,
\end{equation}
\tag{2.10}
$$
where $B \colon \mathbb{Z} \to \mathbb{Z}$ is an arbitrary bijection. Indeed, each of the series in (2.9) and (2.10) is obtained by permuting the terms of the other. The first of them converges absolutely for all $h \ne 0$, since its terms are $o(n^{-2})$. Hence, the second one absolutely converges too, and the sums of both these series coincide. Moreover, combining (2.7) and (2.8), we get that (2.9) is equivalent to $D^2_{\mathrm{symm}} F(x_0)=A$. Thus, the following statement is true. Proposition 1. Suppose we are given $x_0 \in \mathbb{T}$, $A \in \mathbb{C}$, a series (2.1) with coefficients tending to zero, and an odd bijection $B \colon \mathbb{Z} \to \mathbb{Z}$. Then the relations (2.9), (2.10) and $D^2_{\mathrm{symm}} F(x_0)=A$ are equivalent. 2.3. Auxiliary propositionsFor a given $M=2N+1$ consider the $M \times M$ matrix
$$
\begin{equation}
A=(a_{jm})_{-N \leqslant j,m \leqslant N}, \qquad a_{jm} := \exp \biggl(\frac{2 \pi i j m}{M}\biggr).
\end{equation}
\tag{2.11}
$$
This matrix is the well-known discrete Fourier transform matrix, or, in other words, the IDFT-matrix, multiplied by $M$. $A$ is invertible and the bar means complex conjugation (see, for example, [27], [28]). Further, for any integer $Q$, the equality holds:
$$
\begin{equation}
\sum_{j=- N}^{N}\exp\biggl(\frac{2 \pi i Q j}{M}\biggr)=\begin{cases} M &\text{if }Q=0 \ (\operatorname{mod} M), \\ 0 & \text{else.} \end{cases}
\end{equation}
\tag{2.13}
$$
The equality (2.13) is actually a Weil sum (see, for example, [29], Lemma 7), and its value is found as the sum of a finite geometric progression if $Q \ne 0 \ (\operatorname{mod} M)$. The following theorem plays one of the key roles in this work. It is a modification of part of Theorem 1 in [8]. We present the proof for the sake of completeness. Theorem 1. Suppose we are given $N\,{\in}\, \mathbb{N} \cup \{0\}$, $M := 2N +1$, $0\,{<}\,h\,{<}\,\pi/(2M)$, and intervals Proof. Applying (2.7) to each interval $I_j$, we obtain a system of $M$ equations:
$$
\begin{equation}
\sum_{n\in\mathbb{Z}}c_n \exp\biggl( \frac{2 \pi in j}{M}\biggr) \operatorname{sinc}^2(nh)= \frac{\Delta^2 F(I_j)}{4h^2}, \qquad j=- N, \dots, N.
\end{equation}
\tag{2.16}
$$
Rewrite (2.16) as
$$
\begin{equation}
\begin{aligned} \, &\sum_{n=-N}^{N} \exp\biggl( \frac{2 \pi i n j}{M}\biggr) c_n \operatorname{sinc}^2(nh) \nonumber \\ &\ =-\sum_{|n|>N}\exp\biggl( \frac{2 \pi i n j}{M}\biggr) c_n \operatorname{sinc}^2(nh) +\frac{\Delta^2 F(I_j)}{4h^2}, \qquad j=-N, \dots, N. \end{aligned}
\end{equation}
\tag{2.17}
$$
Consider the $M \times M$ matrix $A$ from (2.11) as well as $M \times 1$ column matrices $B=\{b_j\}$, $B'=\{b_j'\}$ and $X=\{x_m\}$ by setting
$$
\begin{equation*}
\begin{aligned} \, b_j &:= -\sum_{|n|>N} \exp\biggl( \frac{2 \pi i n j}{M}\biggr) c_n \operatorname{sinc}^2(nh), \qquad b_j'=\frac{\Delta^2 F(I_j)}{4h^2}, \\ x_m &:= c_m \operatorname{sinc}^2(mh), \qquad j,m=-N, \dots, N. \end{aligned}
\end{equation*}
\notag
$$
The system in (2.17) can be written as $AX=(B+B')$. Applying (2.12) to the matrix $A$, we get the equality $X=Y+Y'$, where
$$
\begin{equation*}
Y=\{y_m\}_{m=-N}^N,\quad Y'=\{y_m'\}_{m=-N}^N, \qquad Y := \frac{1}{M} \, \overline{A} B, \qquad Y' := \frac{1}{M} \, \overline{A} B'.
\end{equation*}
\notag
$$
Using the last formulas, we find $y_m$, $y_m^\prime$ and then $x_m$:
$$
\begin{equation}
\begin{aligned} \, y_m &= \frac{1}{M} \sum_{j=-N}^{N} \overline{a_{mj}} \, b_j= -\frac{1}{M} \sum_{j=-N}^{N} \sum_{|n|>N}\exp\biggl( \frac{2 \pi i(n-m) j}{M}\biggr) c_n \operatorname{sinc}^2(nh) \nonumber \\ &=- \frac{1}{M}\sum_{|n|>N} c_n \operatorname{sinc}^2(nh) \sum_{j=-N}^{N} \exp\biggl( \frac{2 \pi i(n-m) j}{M}\biggr) \nonumber \\ &\!\!\!\!\stackrel{(2.13)}{=} -\sum\nolimits^{\prime} c_n \operatorname{sinc}^2(nh), \nonumber \\ y'_m &= \frac{1}{M} \sum_{j=-N}^{N} \overline{a_{mj}} \, b'_j= \frac{1}{M} \sum_{j=-N}^{N} \exp \biggl( \frac{2 \pi i m j}{M}\biggr) \frac{\Delta^2 F(I_j)}{4h^2}, \nonumber \\ x_m &=- \sum\nolimits^{\prime} c_n \operatorname{sinc}^2(nh) +\frac{1}{M} \sum_{j=-N}^{N} \exp\biggl( \frac{2 \pi i m j}{M}\biggr) \frac{\Delta^2 F(I_j)}{4h^2}. \end{aligned}
\end{equation}
\tag{2.18}
$$
Since $x_m=c_m \operatorname{sinc}^2(mh)$, dividing both parts of (2.18) by $\operatorname{sinc}^2(mh)$, we arrive at (2.14). The proof is completed. The following inequality is well known (see, for example, the proof of Theorem 1.1 in [26]):
$$
\begin{equation}
\int_{-\infty}^\infty \biggl| \frac{d}{dx} \bigl(\operatorname{sinc}^2(x) \bigr) \biggr| \, dx =: I< \infty.
\end{equation}
\tag{2.19}
$$
The following Proposition 2 is an analogue for rearranged trigonometric series of the Cantor–Lebesgue theorem. Note that the permutation $B$ of the set $\mathbb{Z}$ in it is not assumed to be odd. In the proof, we follow, to a large extent, the plan from the proof of Lemma 2.4 in [16]. Proposition 2. Let $\{c_n\}_{n \in \mathbb{Z}}$ be a two-sided sequence of complex numbers, $E \subset \mathbb{T}$ be a set of positive measure, and $B \colon \mathbb{Z} \to \mathbb{Z}$ be a permutation of the set $\mathbb{Z}$, satisfying the condition Proof. We have
$$
\begin{equation*}
\begin{gathered} \, C_n(x) = \biggl[ \exp \biggl(\frac{i B(n) x+i B(-n) x}{2} \biggr) \biggr] A_n(x), \\ A_n(x) := c_{B(n)} \exp \biggl(\frac{i Q(n) x}{2} \biggr)+c_{B(-n)} \exp \biggl(-\frac{i Q(n) x}{2} \biggr). \end{gathered}
\end{equation*}
\notag
$$
For each $x\in E$ from the equality $\lim_{n \to \infty} C_n(x)=0$ it follows that $\lim_{n \to \infty} A_n(x)=0$. By Egorov’s theorem, $A_n(x)$ uniformly converges to zero on some set $E_0 \subset E$ of positive measure. Take into account the fact that the Fourier coefficients of the characteristic function of the set $E_0$ tend to zero. From here and from (2.20) we get
It follows from the above that if $\varepsilon>0$ and $n$ is sufficiently large, then
$$
\begin{equation*}
\begin{aligned} \, \varepsilon^2 |E_0| &\geqslant \int_{E_0} |A_n(x)|^2\, dx = \int_{E_0} \bigl( |c_{B(n)}|^2+|c_{B(-n)}|^2+c_{B(n)} \overline{c_{B(-n)}} \exp(i Q(n) x) \\ &\qquad +\overline{c_{B(n)}} c_{B(-n)} \exp(- i Q(n) x) \bigr)\, dx \\ &\geqslant |E_0| (|c_{B(n)}|^2+|c_{B(-n)}|^2)-|c_{B(n)}| |c_{B(-n)}| (|{\operatorname{Int}_n^{+}}| +|{\operatorname{Int}_n^{-}}|) \\ &\geqslant |E_0| (|c_{B(n)}|^2+|c_{B(-n)}|^2)-\frac{1}{2}(|c_{B(n)}|^2+|c_{B(-n)}|^2) (|{\operatorname{Int}_n^{+}}| + |{\operatorname{Int}_n^{-}}|) \\ &\geqslant \frac{|E_0|}{2} (|c_{B(n)}|^2+|c_{B(-n)}|^2). \end{aligned}
\end{equation*}
\notag
$$
This yields the equality $\lim_{n \to \pm \infty} c_{B(n)}=0$, which is equivalent to (2.21).
It remains to note that $C_n(x)$ is the difference of two partial sums of the series (2.2) with neighboring numbers. Therefore, if such a series converges to a finite sum at each point of the set $E$, then $\lim_{n \to \infty} C_n(x)=0$ whenever $x \in E$. Hence, its coefficients tend to zero, according to proven formula (2.21). This completes the proof. § 3. $D$-monotonicity of one-to-one mappings from $\mathbb{Z}$ to $\mathbb{Z}$Here we assume to be given some odd one-to-one mapping $B \colon \mathbb{Z} \to \mathbb{Z}$ extended to the mapping $B(x) \colon \mathbb{R} \to \mathbb{R}$ in such a way that $B(x)$ is a linear function on each interval $[n, n+1]$, $n \in \mathbb{Z}$. Definition 1. Given $D \in \mathbb{N}$, we say that the mapping $B(n)$ is $D$-monotone if for every non-integer $y \in \mathbb{R}$ the equation $B(x)=y$ has at most $D$ solutions. It can be shown that if we consider all $y \in \mathbb{R}$ in Definition 1, including integers, we get an equivalent definition, but we will not need it. It is clear that if the function $f \colon \mathbb{R} \to \mathbb{R}$ is continuous and for all real $y$ the equation $B(x)=y$ has at most one solution, then $f$ is strictly monotonous. From this point of view, the $D$-monotonicity of the mapping $B$ means in a certain sense that $B$ “not too often” changes the direction of monotonicity (the “frequency” depends on $D$). Notice, that there are only two $1$-monotone mappings: the identity mapping $\mathrm{Id} \colon \mathbb{Z} \to \mathbb{Z}$ and $-\mathrm{Id}$. For each $n \in \mathbb{Z}$ denote by $I_n$ an open interval with endpoints $B(n)$ and $B(n+1)$:
$$
\begin{equation}
I_n = \bigl( \min \{B(n), B(n+1)\}, \, \max \{B(n), B(n+1)\} \bigr).
\end{equation}
\tag{3.1}
$$
Now we can say that the $D$-monotonicity of $B$ is equivalent to the fact that each non-integer $y \in \mathbb{R}$ is an element of at most $D$ intervals of the form $I_n$. We denote by $\mathcal{B}_D$ the class consisting of all $D$-monotone bijections, and $\mathcal{B}_{\mathcal{M}} := \bigcup_{D \in \mathbb{N}} \mathcal{B}_D$. Lemma 1. If $B \in \mathcal{B}_D$ and a function $f$ is absolutely integrable on $\mathbb{R}$, then inequality holds for all $h >0$. Proof. Set $J_m := (m, m+1)$, $m \in \mathbb{Z}$. It follows from Definition 1 that, for each $m$, any point $x \in J_m$ belongs to at most $D$ intervals of the form $I_n$. The endpoints of the interval $I_n$ are integers, and the endpoints of the interval $J_m$ are neighboring integers. So either $I_n \cap J_m=\varnothing$ or $I_n \supset J_m$. Hence, each $J_m$ is a subset of at most $D$ intervals $I_n$. Therefore, for all $m \in \mathbb{Z}$ and $h >0$ the set $h J_m$ lies in at most $D$ intervals $h I_n$. Then
$$
\begin{equation*}
\begin{aligned} \, \sum_{n \in \mathbb{Z}} \int_{h I_n} |f(x)| \, dx &= \sum_{n \in \mathbb{Z}} \sum_{J_m \subset I_n} \int_{h J_m} |f(x)| \, dx \\ &\leqslant D \sum_{m \in \mathbb{Z}} \int_{h J_m} |f(x)| \, dx = D \int_{\mathbb{R}} |f(x)|\, dx. \end{aligned}
\end{equation*}
\notag
$$
The Lemma is proved. Theorem 2. Let $B \in \mathcal{B}_{\mathcal{M}}$. If the series (2.2) with coefficients tending to zero converges to a finite sum $A$ at a point $x_0$, then (2.10) holds. Proof. Since $B \in \mathcal{B}_{\mathcal{M}}$, we have $B \in \mathcal{B}_D$ for some natural $D$. The further proof basically repeats the reasoning in the classical situation when $B=\mathrm{Id}$ (see, for example, [5], Ch. 1, § 68). For $n \in \mathbb{N} \cup \{0\}$ and $h>0$ we put
$$
\begin{equation*}
\begin{gathered} \, C_n := [ c_{B(n)} \exp(i B(n) x_0)+ c_{-B(n)} \exp(- i B(n) x_0)] \cdot \begin{cases} 1 &\text{if }n \in \mathbb{N}, \\ \dfrac12 &\text{if }n=0, \end{cases} \\ r_n := \sum_{k=n+1}^\infty C_k, \qquad C_n(h)=C_n \operatorname{sinc}^2(B(n) h). \end{gathered}
\end{equation*}
\notag
$$
Taking into account the oddness of the bijection $B$, it can be seen that the expression in brackets in the formula (2.10) is $\sum_{n=0}^\infty C_n(h)$. We have
Take an arbitrary $\varepsilon >0$. First, the series (2.2) converges to a finite sum $A$ at the point $x_0$. Therefore, there exists $N$ such that $|r_n|<\varepsilon$ for all $n \geqslant N$ and the second term on the right-hand side of (3.3) is less than $\varepsilon$. Second, for every fixed $n$ we have $\operatorname{sinc}^2(B(n) h) \to 1$ as $h \to 0$. Hence, if $h$ is small enough, then the first term on the right-hand side of (3.3) is also less than $\varepsilon$. Thirdly, $B \in \mathcal{B}_{\mathcal{M}}$ implies $B \in \mathcal{B}_D$ for some $D \in \mathbb{N}$. We have
$$
\begin{equation}
\begin{aligned} \, &\biggl| \sum_{n=N+1}^\infty C_n(h)\biggr| = \biggl| \sum_{n=N+1}^\infty (r_n-r_{n -1}) \operatorname{sinc}^2(B(n) h)\biggr| \nonumber \\ &\ =\biggl| -r_N \operatorname{sinc}^2(B(N+1) h) + \sum_{n=N+1}^\infty r_n [\operatorname{sinc}^2(B(n) h)-\operatorname{sinc}^2(B(n+1) h)] \biggr| \nonumber \\ &\ =\biggl| -r_N \operatorname{sinc}^2(B(N+1) h) -\sum_{n=N+1}^\infty r_n \int_{B(n) h}^{B(n+1) h} \frac{d}{dx} (\operatorname{sinc}^2 x) \, dx \biggr| \nonumber \\ &\leqslant \varepsilon + \varepsilon \sum_{n=N+1}^\infty \biggl| \int_{B(n) h}^{B(n+1) h} \biggl| \frac{d}{dx} (\operatorname{sinc}^2 x) \biggr| \, dx \biggr| \nonumber \\ &\leqslant \varepsilon + \varepsilon \sum_{n \in \mathbb{Z}} \biggl| \int_{B(n) h}^{B(n+1) h} \biggl| \frac{d}{dx} (\operatorname{sinc}^2 x) \biggr| \, dx \biggr| \stackrel{(3.2)}{\leqslant} \varepsilon+\varepsilon D I, \end{aligned}
\end{equation}
\tag{3.4}
$$
where $I$ is defined in (2.19). Since it could turn out that $B(n)h>B(n+ 1) h$, we wrote the modulus sign twice in the final part of the chain (3.4).
Thus, for sufficiently small $h$ the left-hand side of (3.3) does not exceed $3 \varepsilon+\varepsilon D I$. This establishes the formula (2.10) and completes the proof. Theorem 3. If $B \in \mathcal{B}_{\mathcal{M}}$ and the $B$-rearranged trigonometric series $TS$ converges everywhere to a finite integrable function $f$, then $TS$ is the Fourier series of $f$. In other words, the empty set is a $V$-set for the system $\{\exp(i B(n) x)\}_{n \in \mathbb{Z}}$ whenever $B \in \mathcal{B}_{\mathcal{M}}$. Proof. By assumption, the $B$-rearranged trigonometric series $TS$ converges everywhere to a finite integrable function $f$. Proposition 2 yields that the coefficients of $TS$ tend to zero.
Now let us apply Theorem 2. We obtain that at each point $x_0 \in(-\pi, \pi)$ relation (2.10) with $A=f(x_0)$ holds. This implies together with Proposition 1 that $D^2_{\mathrm{symm}} F(x)=f(x)$ for all $x \in(-\pi, \pi)$, where $F$ is the Riemann function of $TS$. Notice that $F$ is continuous on $[-\pi, \pi]$ and use Theorem A1. We get the identity
$$
\begin{equation*}
F(x) \equiv \int_{-\pi}^x dt \int_{-\pi}^t f(u) \, du + Ax+B,\qquad A, B\ \unicode{x2013}\ \text{const}.
\end{equation*}
\notag
$$
From the last formula by standard reasoning (see, for example, [4], Ch. 11, 11.3) we obtain that $TS$ is the Fourier series of the function $f$. The theorem is proved. § 4. Constructions of special subsets of $\mathbb{T}$ and rearrangements of the trigonometric system4.1. Sets of Rajchman–Zygmund typeHere sets will be constructed that have a certain symmetry inside the group $\mathbb{T}$. Similar constructions can be found in Rajchman and Zygmund works (see, for example, [5], Ch. 14, § 7, § 26). Take any increasing sequence of odd numbers $\mathbf{M}=\{M(s)\}_{s=1}^\infty$ and a sequence of positive numbers tending to zero $\mathbf{h}=\{h(s)\}_{s=1}^\infty$ such that
$$
\begin{equation}
\text{the numbers }\frac{M(s+1)}{M(s)}\text{ are integer (odd) for all }s =0, 1, \dots,
\end{equation}
\tag{4.3}
$$
$$
\begin{equation}
\sup_{s \in \mathbb{N}} \frac{1}{h^2(s) M(s)(N(s+1)-M(s))} =: E<\infty.
\end{equation}
\tag{4.4}
$$
Here $\{N(s)\}$ are integers related to $M(s)$ by Relations (4.1)–(4.4) are true if, for example, the number $p \geqslant 5$ is odd and Indeed, it is easy to verify that the given sequence satisfies (4.1)–(4.3). Let us check that the relation (4.4) holds:
$$
\begin{equation*}
\begin{aligned} \, &\frac{1}{h^2(s) M(s)(N(s+1)-M(s))} = \frac{p^{2 s^2+2 s}}{p^{s^2}((p^{(s+1)^2}-1)/2-p^{s^2})} \\ &\qquad\leqslant \frac{p^{2 s^2+2 s}}{p^{s^2}(p^{(s+1)^2}/3-p^{s^2})} = \frac{3 p^{2 s}}{p^{2 s+1}-3} = \frac{3}{p-3/p^{2 s}} \leqslant \frac{3}{4}, \end{aligned}
\end{equation*}
\notag
$$
since $p \geqslant 5$ and $s \geqslant 1$. So, (4.4) is true if $E=3/4$. Further, the sets and their elements are considered here as subsets and elements of the group $\mathbb{T}$, not $\mathbb{R}$. With each pair $(\mathbf{M}, \mathbf{h})$ satisfying conditions (4.1)–(4.4), we associate a countable set $H=H(\mathbf{M}, \mathbf{h}) \subset \mathbb{T}$ as well as open sets $G^q=G^q(\mathbf{M}, \mathbf{h}) \subset \mathbb{T}$, where $q \in \mathbb{N}$. To do this, for each $s$ consider an $M(s)$-periodic set $\{x_{s,j}\}_{j \in \mathbb{Z}} \subset \mathbb{T}$ which is the centered uniform grid with $M(s)$ nodes:
$$
\begin{equation}
x_{s,j} := \frac{2 \pi j}{M(s)},\qquad j \in \mathbb{Z}.
\end{equation}
\tag{4.6}
$$
Then, for each node, we construct an interval $I_{s,j}$ of length $2 h(s)$ centered at this node:
$$
\begin{equation}
I_{s,j} :=(x_{s,j}-h(s), \, x_{s,j}+h(s)),\qquad j \in \mathbb{Z}.
\end{equation}
\tag{4.7}
$$
Finally, set
$$
\begin{equation}
H:= \bigcup_{s=1}^\infty \bigcup_{j \in \mathbb{Z}}\{x_{s,j}\}, \qquad G^q := \bigcup_{s=q}^\infty \bigcup_{j \in \mathbb{Z}} I_{s,j}, \quad q \in \mathbb{N}.
\end{equation}
\tag{4.8}
$$
Notice that
$$
\begin{equation}
I_{s,j}= x_{s,j} \oplus I_{s,0},\qquad s \in \mathbb{N}, \quad j \in \mathbb{Z}.
\end{equation}
\tag{4.9}
$$
For any fixed $s$ the sequence $\{x_{s,j}\}_{j \in \mathbb{Z}}$ is $M(s)$-periodic. Therefore, (4.9) implies $M(s)$-periodicity of the interval sequence $\{I_{s,j}\}_{j \in \mathbb{Z}}$. In this connection we can write $j \in \{-N(s), \dots, N(s)\}$ instead $j \in \mathbb{Z}$ in formulas (4.6)–(4.8). Lemma 2. The following are true. (A) For $t \leqslant s$ the number $M(s) / M(t)$ is an integer and for all integers $j$ and $k$.(B) If $t \leqslant s$, then for all integers $j$ and $k$ (C) $H$ is a subgroup of the group $\mathbb{T}$. Proof. (A) The condition (4.3) implies that $M(s) / M(t)$ if $t \leqslant s$. Next, which proves (A).
(B) We have
$$
\begin{equation*}
x_{t, k} \oplus I_{s, j} \stackrel{(4.9)}{=} x_{t, k} \oplus x_{s, j} \oplus I_{s, 0} \stackrel{(4.10)}{=} x_{s, j+k M(s)/M(t)} \oplus I_{s, 0} \stackrel{(4.9)}{=} I_{s, j+k M(s)/M(t)}.
\end{equation*}
\notag
$$
The statement (B) is proved.
(C) From (4.8) we see that $0 \in H$, and that $-h \in H$ if $h \in H$. Let now $t \leqslant s$, $a=x_{t,k} \in H$, $b=x_{s,j} \in H$. According to (4.6) $a \oplus b=x_{s, j+k M(s)/M(t)}$, whence $a \oplus b \in H$. This proves the statement (C). Lemma 3. For any $\delta>0$ there exists $q \in \mathbb{N}$ such that $|\mathbb{T} \setminus G^q| >2 \pi-\delta$. Proof. Let us estimate the measure of the set $G^q$:
$$
\begin{equation*}
|G^q| < \sum_{s=q}^\infty \sum_{j=-N(s)}^{N(s)} 2 h(s) = 2 \sum_{s=q}^\infty M(s) h(s).
\end{equation*}
\notag
$$
By (4.2), we have $\sum_{s=q}^\infty M(s) h(s)<\delta / 2$ for sufficiently large $q$. For the same $q$ the inequality $|G^q|<\delta$ is true, whence $|\mathbb{T} \setminus G^q| >2 \pi-\delta$. This completes the proof. 4.2. Rearrangements of the trigonometric systemTo each pair of sequences $(\mathbf{M}, \mathbf{h})$ satisfying (4.1)–(4.4), we assign an odd permutation $\widehat{B}=\widehat{B}(\mathbf{M}, \mathbf{h})$ of the set $\mathbb{Z}$. To do this, first we split the set $\mathbb{N} \cup \{0\}$ into successive pairwise disjoint sets $\{0, \dots, N(0)\}$ and $\operatorname{Block}(s)$ ($s \in \mathbb{N}$):
$$
\begin{equation}
\begin{gathered} \, \mathbb{N} \cup \{0\} = \{0, \dots, N(0)\} \bigsqcup \biggl( \bigsqcup_{s=1}^\infty \operatorname{Block}(s)\biggr), \\ \operatorname{Block}(s) := \{n \in \mathbb{N} \colon N(s)+1 \leqslant n \leqslant N(s+1)\}. \nonumber \end{gathered}
\end{equation}
\tag{4.12}
$$
With each set (block) $\operatorname{Block}(s)$ we associate a non-negative integer $m_s$ in such a way that $m_s \leqslant N(s)$ and
$$
\begin{equation}
\begin{aligned} \, &\text{each }m \in \mathbb{N} \cup \{0\}\text{ occurs in the sequence }\{m_s\} \\ &\text{infinitely many times}. \end{aligned}
\end{equation}
\tag{4.13}
$$
Now consider the sets
$$
\begin{equation}
\operatorname{Progr}(s) := \{ n \in \operatorname{Block}(s) \colon n=\pm m_s \ (\operatorname{mod} M(s))\},
\end{equation}
\tag{4.14}
$$
$$
\begin{equation}
\operatorname{Segm}(s) := \{n \in \mathbb{N} \colon N(s)+1 \leqslant n \leqslant N(s)+\# \operatorname{Progr}(s)\}.
\end{equation}
\tag{4.15}
$$
Each set $\operatorname{Segm}(s)$ is a segment in $\mathbb{N}$, i. e. it is finite and consists of consecutive natural numbers. The sets $\operatorname{Progr}(s)$ are one (when $m_s=0$) or the union of two (when $m_s \ne 0$) finite arithmetic progressions with the same difference $M(s)$. It yields from (4.15) that
$$
\begin{equation*}
\# \operatorname{Segm}(s)=\# \operatorname{Progr}(s).
\end{equation*}
\notag
$$
Besides,
$$
\begin{equation}
\operatorname{Progr}(s) \subset \operatorname{Block}(s), \qquad \operatorname{Segm}(s) \subset \operatorname{Block}(s).
\end{equation}
\tag{4.16}
$$
The first inclusion in (4.16) immediately follows from (4.14). To establish the second one, we note the both sets $\operatorname{Segm}(s)$ and $\operatorname{Block}(s)$ are segments in $\mathbb{N}$ with the same left endpoints and
$$
\begin{equation*}
\# \operatorname{Segm}(s)=\# \operatorname{Progr}(s) \leqslant \# \operatorname{Block}(s)
\end{equation*}
\notag
$$
(the last inequality holds because $\operatorname{Progr}(s) \subset \operatorname{Block}(s)$). So $\operatorname{Segm}(s) \subset \operatorname{Block}(s)$. It can be seen from the previous constructions that there exists a (unique!) bijection $\widehat{B} \colon \mathbb{Z} \to \mathbb{Z}$ such that:
$$
\begin{equation}
\widehat{B} (\operatorname{Segm}(s)) = \operatorname{Progr}(s),
\end{equation}
\tag{4.18}
$$
$$
\begin{equation}
\widehat{B}(n_1) < \widehat{B}(n_2), \quad \text{if }n_1<n_2\text{ and }n_1,n_2 \in \operatorname{Segm}(s),
\end{equation}
\tag{4.19}
$$
$$
\begin{equation}
\widehat{B} (\operatorname{Block}(s) \setminus \operatorname{Segm}(s)) = \operatorname{Block}(s) \setminus \operatorname{Progr}(s),
\end{equation}
\tag{4.20}
$$
$$
\begin{equation}
\widehat{B}(n_1) < \widehat{B}(n_2), \quad \text{if }n_1<n_2\text{ and }n_1,n_2 \in \operatorname{Block}(s) \setminus \operatorname{Segm}(s), \quad s \in \mathbb{N};
\end{equation}
\tag{4.21}
$$
$$
\begin{equation}
\widehat{B}(-n)=- \widehat{B}(n), \qquad n \in \mathbb{N} \cup \{0\}.
\end{equation}
\tag{4.22}
$$
In the following lemma, two properties of the bijection $\widehat{B}$ are noted, which immediately follow from (4.17)–(4.21). Lemma 4. 1) The mapping $\widehat{B}$ preserves the set $\{0, \dots, N(0)\}$, being identical upon restricted to it. 2) $\widehat{B}$ preserves all blocks $\operatorname{Block}(s)$, and on each block $\operatorname{Block}(s)$ the bijection behaves as follows: first the sequence $\widehat{B}(n)$ increases on the set $\operatorname{Segm}(s);$ then decreases when $n$ is the last point of the set $\operatorname{Segm}(s)$ and $n+1$ is the first point of the set $\operatorname{Block}(s) \setminus \operatorname{Segm}(s);$ finally increases again on the set $\operatorname{Block}(s) \setminus \operatorname{Segm}(s)$. Lemma 5. $\widehat{B} \in \mathcal{B}_3$ and, as a corollary, $\widehat{B} \in \mathcal{B}_{\mathcal{M}}$. Proof. Consider an arbitrary non-integer $x$ and intervals $I_n$ in (3.1) with $B := \widehat{B}$. It is necessary to prove that the relation $x \in I_n$ holds for at most three $I_n$. Taking (4.22) into account, it suffices to consider the case $x>0$ and $n \in \mathbb{N} \cup \{0\}$.
Let, first, $0< x<N(0)$. Lemma 4, 1) implies that the relation $x \in I_n$ can hold only for one $I_n$, namely, for $I_n=(\widehat{B}(0), \widehat{B}(1))$. Let now $x \in \operatorname{Block}(s)$ for some $s$, i. e. $N(s)+1<x<N(s+1)$ (let us keep in mind that $x$ is a non-integer). Then it follows from Lemma 4, 2) that the relation $x \in I_n$ can hold at most once in each of the following situations:
$$
\begin{equation*}
\begin{gathered} \, \begin{alignedat}{2} I_n &=\bigl(\widehat{B}(n), \widehat{B}(n+1) \bigr), &\qquad n &\in \operatorname{Segm}(s), \\ I_n &=\bigl(\widehat{B}(n), \widehat{B}(n+1) \bigr), &\qquad n &\in \operatorname{Block}(s) \setminus \operatorname{Segm}(s); \end{alignedat} \\ \begin{aligned} \, I_n=\bigl(\widehat{B}(n+1), \widehat{B}(n) \bigr), \quad &\text{where }n\text{ is the last element } \operatorname{Segm}(s), \\ &n+1\text{ is the first element }\operatorname{Block}(s) \setminus \operatorname{Segm}(s). \end{aligned} \end{gathered}
\end{equation*}
\notag
$$
The statement of the Lemma follows from what has been said. § 5. Main resultsIn Theorems 4–9 we will assume that arbitrary sequences $\mathbf{M}=\{M(s)\}$, $\mathbf{h}=\{h(s)\}$, and $\{N(s)\}$, satisfying (4.1)–(4.5) are chosen and fixed. We will consider the countable set $H=H(\mathbf{M}, \mathbf{h})$ and the open sets $G^q=G^q(\mathbf{M}, \mathbf{h})$, $q \in \mathbb{N}$, constructed in (4.8), as well as the bijection $\widehat{B}=\widehat{B}(\mathbf{M}, \mathbf{h})$, introduced in subsection 4.2. Recall that
$$
\begin{equation*}
\begin{gathered} \, x_{s,j}=\frac{2 \pi j}{M(s)}, \qquad I_{s,j}=(x_{s,j}-h(s), \, x_{s,j}+h(s)), \\ s \in \mathbb{N},\qquad j=- N(s), \dots, N(s). \end{gathered}
\end{equation*}
\notag
$$
Theorem 4. Let $TS$ be a trigonometric series of the form (2.1), satisfying the condition (2.21). Assume that the series $\widehat{B} TS$, which is the $\widehat{B}$-rearrangement of series $TS$, converges to a finite sum at each point $x \in H$. Then for all $m \in \mathbb{Z}$ we have Proof. The formula (5.1) is invariant under the replacement of $m$ by ${-}m$, hence, to prove it, it suffices to confine the case $m \in \mathbb{N} \cup \{0\}$. We fix an arbitrary $m \in \mathbb{N} \cup \{0\}$. Since $0 \in H$, it follows from the conditions of the Theorem that the series $\widehat{B} TS$ converges to a finite sum at the point $x=0$. Therefore, the value $\sigma_n := \sum_{|k| \geqslant n+1} c_{\widehat{B}(k)}$, $n \in \mathbb{N}$, which is the $n$ th remainder of the series $\widehat{B} TS$ at the point $x=0$, is well defined and tends to zero as $n \to \infty$.
We take an arbitrary $\varepsilon>0$. It follows from $\lim_{n \to \infty} \sigma_n=0$ and (2.21) that
$$
\begin{equation}
|\sigma_n|<\varepsilon \quad\text{and }\quad |c_n|<\varepsilon \quad\text{for sufficiently large }s.
\end{equation}
\tag{5.3}
$$
Consider an arbitrary sufficiently large $s$ satisfying (5.2). Since $\operatorname{sinc}^2(x) \to 1$ for $x \to 0$, and $h(s) \to 0$ for $s \to \infty$ (see (4.2)), then we can additionally assume that
$$
\begin{equation}
\biggl| \operatorname{sinc}^{-2} \biggl(\frac{m h(s)}{2} \biggr) \biggr| \leqslant 2.
\end{equation}
\tag{5.4}
$$
Put
$$
\begin{equation}
M := M(s),\qquad N := N(s), \qquad N^{+} := N(s+1), \qquad h := \frac{h(s)}{2}.
\end{equation}
\tag{5.5}
$$
Compute $c_m+c_{-m}$ using Theorem 1, taking $M$, $N$ and $h$ from (5.5), and, also, $I_j := I_{s,j}$. We get (2.14) and (2.15) that
$$
\begin{equation}
\begin{aligned} \, &c_m+c_{-m} = -\sum\nolimits^{\prime\prime} c_n \, \frac{\operatorname{sinc}^2(nh)}{\operatorname{sinc}^2(mh)} \nonumber \\ &\ \qquad +\frac{1}{4 M h^2 \operatorname{sinc}^2(mh)} \sum_{j=- N}^{N} \biggl[ \exp\biggl( \frac{2 \pi i m j}{M}\biggr)+ \exp\biggl(-\frac{2 \pi i m j}{M} \biggr) \biggr] \Delta^2 F(I_{s,j}) \nonumber \\ &\ =-\sum\nolimits^{\prime\prime} c_n \, \frac{\operatorname{sinc}^2(nh)}{\operatorname{sinc}^2(mh)} +\frac{1}{2 M h^2 \operatorname{sinc}^2(mh)}\sum_{j=- N}^{N} \cos(m x_{s,j}) \Delta^2 F(I_{s,j}). \end{aligned}
\end{equation}
\tag{5.6}
$$
Here $\sum\nolimits^{\prime\prime}$ extends to all $n$ belonging to the set
$$
\begin{equation}
\{n \colon |n|>N \text{ and } n=\pm m \ (\operatorname{mod} M)\}.
\end{equation}
\tag{5.7}
$$
Let us estimate the expression $\sum\nolimits^{\prime\prime} c_n \, (\operatorname{sinc}^2(nh)/\operatorname{sinc}^2(mh))$ and show that its absolute value is sufficiently small. Looking at (4.12), (4.14), (5.2) and (5.7), we conclude that the set mentioned in (5.7) is $A_1 \bigsqcup A_2$, where
$$
\begin{equation}
\begin{aligned} \, A_1 &:= \{n \colon N<|n| \leqslant N^{+}\text{ and } n=\pm m \ (\operatorname{mod} M)\} = \{n \colon |n| \in \operatorname{Progr}(s)\}, \\ A_2 &:= \{n \colon |n|>N^{+} \text{ and } n=\pm m \ (\operatorname{mod} M)\}. \end{aligned}
\end{equation}
\tag{5.8}
$$
So, $\sum\nolimits^{\prime\prime} c_n (\operatorname{sinc}^2(nh)/\operatorname{sinc}^2(mh))= \sum\nolimits^{\prime\prime}_1 + \sum\nolimits^{\prime\prime}_2$, where the sum $\sum\nolimits^{\prime\prime}_1$ applies to $n \in A_1$ whereas $\sum\nolimits^{\prime\prime}_2$ applies to $n \in A_2$.
We will see that the sum $\sum\nolimits^{\prime\prime}_1$ is small because the series $\widehat{B} TS$ converges to a finite sum at the point $x=0$, and therefore the first inequality in (5.3) is true. The sum $\sum\nolimits^{\prime\prime}_2$ is also small because the quantities $n$ are sufficiently large whereas the value $h=h(s)/2$ is not too small due to the constraint in (4.4); as a consequence, the values $\operatorname{sinc}^2(nh)$ in this sum turn out to be sufficiently small. First, let us estimate $\sum\nolimits^{\prime\prime}_1$. Initially, this sum consists of some number of “scattered” members of the series $TS$, multiplied by $\operatorname{sinc}^2(n h)/\operatorname{sinc}^2(m h)$, and in the general case it can be big. But if we rearrange these terms, collecting them together consecutively, then, due to the convergence to zero at the point $x=0$ of the $\widehat{B}$-rearranged series $TS$, the sum $\sum\nolimits^{\prime\prime}_1$ becomes small. This moment is the keystone in the proof of the Theorem. We have
$$
\begin{equation}
\begin{aligned} \, \biggl|\sum\nolimits^{\prime\prime}_1\biggr| &= |{\operatorname{sinc}^{-2}(m h)}| \biggl| \sum_{|n| \in \operatorname{Progr}(s)} c_n \operatorname{sinc}^2(n h) \biggr| \nonumber \\ &\stackrel{(5.4)}{\leqslant} 2\biggl| \sum_{|n| \in \operatorname{Progr}(s)} c_n \operatorname{sinc}^2(n h)\biggr| \stackrel{(4.18)}{=} 2\biggl| \sum_{|n| \in \operatorname{Segm}(s)} c_{\widehat{B}(n)} \operatorname{sinc}^2 \bigl(\widehat{B}(n) h \bigr) \biggr| \nonumber \\ &\!\!\!\!\!\!\!\!\!\stackrel{(4.15), \, (4.22)}{=} 2 \biggl| \sum_{n=Q}^{P} \bigl[ c_{\widehat{B}(n)}+c_{-\widehat{B}(n)} \bigr] \operatorname{sinc}^2 \bigl(\widehat{B}(n) h \bigr) \biggr|, \end{aligned}
\end{equation}
\tag{5.9}
$$
where $Q$ is the smallest and $P$ is the largest number from the set $\operatorname{Segm}(s)$. Remembering the notation $\sigma_n := \sum_{|k|=n+1}^\infty c_{\widehat{B}(k)}$, we apply the Abel transform and continue chain (5.9):
$$
\begin{equation}
\begin{aligned} \, \Bigl| \sum\nolimits^{\prime\prime}_1 \Bigr| &\leqslant 2\biggl| \sum_{n=Q}^{P} (\sigma_{n-1}-\sigma_n) \operatorname{sinc}^2 \bigl(\widehat{B}(n) h \bigr) \biggr| \leqslant 2\bigl|\sigma_{Q-1} \operatorname{sinc}^2 \bigl(\widehat{B}(Q) h \bigr)\biggr| \nonumber \\ &\qquad + 2\bigl| \sigma_{P}\operatorname{sinc}^2 \bigl(\widehat{B}(P) h \bigr)\bigr| +2\biggl| \sum_{n=Q}^{P-1} \sigma_{n} \bigl( \operatorname{sinc}^2 \bigl(\widehat{B}(n+1) h \bigr) - \operatorname{sinc}^2 \bigl(\widehat{B}(n) h \bigr) \bigr) \biggr| \nonumber \\ &\!\!\stackrel{(5.3)}{\leqslant} 4 \varepsilon +2\varepsilon \sum_{n=Q}^{P-1} \biggl| \int_{\widehat{B}(n) h}^{\widehat{B}(n+1) h} \biggl| \frac{d}{dx} (\operatorname{sinc}^2 x) \biggr| \, dx \biggr| \nonumber \\ &\leqslant 4 \varepsilon +2 \varepsilon \sum_{n \in \mathbb{Z}} \biggl| \int_{\widehat{B}(n) h}^{\widehat{B}(n+1) h} \biggl| \frac{d}{dx} (\operatorname{sinc}^2 x) \biggr| \, dx \biggr| \stackrel{(2.19), \, (3.2), \, \text{Lemma }5}{\leqslant} 4 \varepsilon + 6 \varepsilon I. \end{aligned}
\end{equation}
\tag{5.10}
$$
Let us now estimate $\sum\nolimits_2$:
$$
\begin{equation}
\begin{aligned} \, \Bigl| \sum\nolimits_2 \Bigr| &\leqslant |{\operatorname{sinc}^{-2}(mh)}| \sum_{\substack{|n|>N'\\n=\pm m \ (\operatorname{mod} M)}} \frac{|c_n|}{(n h)^2} \nonumber \\ &\!\!\!\!\!\!\!\!\!\stackrel{(5.3), \, (5.4)}{\leqslant} \frac{2 \varepsilon}{h^2} \sum_{\substack{|n|>N'\\n=\pm m \ (\operatorname{mod} M)}} \frac{1}{n^2} = \frac{4 \varepsilon}{h^2} \sum_{\substack{n>N'\\n=\pm m \ (\operatorname{mod} M)}} \frac{1}{n^2}. \end{aligned}
\end{equation}
\tag{5.11}
$$
Take all consecutive numbers $n>N'$ such that $n=\pm m \ (\operatorname{mod} M)$ and split them into pairs. Then the numbers from the first pair are greater than $N'$, from the second are greater than $N'+M$, from the third are greater than $N'+2M$, and so on. Using this observation, we will continue the chain (5.11):
$$
\begin{equation}
\begin{aligned} \, \Bigl| \sum\nolimits_2 \Bigr| &\leqslant \frac{8 \varepsilon}{h^2} \sum_{j=0}^\infty \frac{1}{(N'+j M(s))^2} \leqslant \frac{8 \varepsilon}{h^2} \sum_{j=0}^\infty \int_{j-1}^{j} \frac{dt}{(N'+t M)^2} \nonumber \\ &= \frac{8 \varepsilon}{h^2} \int_{-1}^\infty \frac{dt}{(N'+t M)^2} = \frac{8 \varepsilon}{h^2 M} \int_{N'-M}^\infty \frac{du}{u^2} = \frac{8 \varepsilon}{h^2 M(N'-M)} \nonumber \\ &\!\!\!\stackrel{(5.5)}{=} \frac{32 \varepsilon}{h^2(s) M(s)(N(s+1)-M(s))} \stackrel{(4.4)}{\leqslant} 32 \varepsilon E. \end{aligned}
\end{equation}
\tag{5.12}
$$
Combining (5.10) and (5.12) we have:
$$
\begin{equation}
\biggl| \sum\nolimits^{\prime\prime} c_n \, \frac{\operatorname{sinc}^2(nh)}{\operatorname{sinc}^2(mh)} \biggr| \leqslant \varepsilon (4+6I+32 E).
\end{equation}
\tag{5.13}
$$
Recall again (the formula (5.5)) that $M=M(s)$, $h=h(s)/2$. The formula (5.13) means that the first sum in (5.6) is arbitrarily small for sufficiently large $s$ satisfying the condition (5.2). Passing to the corresponding limit, we obtain (5.1) from (5.6). The theorem is proved. Theorem 5. Under the conditions of Theorem 4, for all $m \in \mathbb{Z}$ we have Proof. We choose an arbitrary $m \in \mathbb{Z}$. Take a natural $t$ so large that $|m| \leqslant N(t)$. According to the conditions of Theorem 4, the series $\sum_{n \in \mathbb{Z}} c_{\widehat{B}(n)} \exp(i \widehat{B}(n) x)$ converges to a finite sum $f(x)$ at each point $x \in H$. Since (Lemma 2, (C)) $H$ is a subgroup of $\mathbb{T}$ and $x_{t,1}=2 \pi/M(t) \in H$, then $H \oplus x_{t,1}=H$. So the series
$$
\begin{equation*}
\sum_{n\in\mathbb{Z}} c_{\widehat{B}(n)} \exp \bigl(i \widehat{B}(n)(x \oplus x_{t,1}) \bigr),
\end{equation*}
\notag
$$
which is a $\widehat{B}$-rearrangement of the series
$$
\begin{equation*}
\widetilde{TS} := \sum_{n\in\mathbb{Z}} \widetilde{c}_n \exp(inx),\qquad \widetilde{c}_n := c_n \exp (i n x_{t,1}),
\end{equation*}
\notag
$$
also converges to a finite sum at every point $x \in H$. Applying Theorem 4 to the series $\widetilde{TS}$, we obtain
$$
\begin{equation}
\begin{aligned} \, \widetilde{c}_m+\widetilde{c}_{-m} &= c_m \exp(i m x_{t,1})+c_{-m} \exp(- i m x_{t,1}) \nonumber \\ &=\lim \frac{2}{M(s) h^2(s)} \sum_{j=- N(s)}^{N(s)} \cos(m x_{s,j}) \Delta^2 \widetilde{F}(I_{s,j}). \end{aligned}
\end{equation}
\tag{5.15}
$$
Here $\widetilde{F}$ is the Riemann function of the series $\widetilde{TS}$, and the limit is understood in the same way as in Theorem 4. Compute $\widetilde{F}$:
$$
\begin{equation*}
\begin{aligned} \, \widetilde{F}(x) &= \frac{\widetilde{c}_0 x^2}{2} - \sum_{n \in \mathbb{Z} \setminus \{0\}} \frac{\widetilde{c}_n}{n^2} \exp(inx) \\ &= \frac{c_0 x^2}{2} - \sum_{n \in \mathbb{Z} \setminus \{0\}} \frac{c_n}{n^2} \exp(i n(x \oplus x_{t,1})) = \widetilde{F}(x \oplus x_{t,1}). \end{aligned}
\end{equation*}
\notag
$$
This yields
$$
\begin{equation}
\Delta^2 \widetilde{F}(I) = \Delta^2 F(x_{t,1} \oplus I) \quad \text{for every interval }I \subset \mathbb{T}.
\end{equation}
\tag{5.16}
$$
The formulas (5.15) and (5.16) give the following:
$$
\begin{equation}
\begin{aligned} \, &c_m \exp(i m x_{t,1}) + c_{-m} \exp(- i m x_{t,1}) \nonumber \\ &\ =\lim \frac{2}{M(s) h^2(s)} \sum_{j=- N(s)}^{N(s)} \cos(m x_{s,j}) \Delta^2 F(x_{t,1} \oplus I_{s,j}) \nonumber \\ &\!\!\stackrel{(4.11)}{=} \lim \frac{2}{M(s) h^2(s)} \sum_{j=- N(s)}^{N(s)} \cos(m x_{s,j}) \Delta^2 F(I_{s, j+M(s)/M(t)}) \nonumber \\ &\ = \lim \frac{2}{M(s) h^2(s)} \sum_{j=- N(s)}^{N(s)} \cos(m x_{s,j-M(s)/M(t)}) \Delta^2 F(I_{s,j}) \nonumber \\ &\ =\lim \frac{2}{M(s) h^2(s)} \sum_{j=- N(s)}^{N(s)} \cos(m x_{s,j}-m x_{t,1}) \Delta^2 F(I_{s,j}) \nonumber \\ &\ =\lim \frac{1}{M(s) h^2(s)} \sum_{j=- N(s)}^{N(s)} [\exp(i m(x_{s,j}-x_{t,1})) + \exp(-i m(x_{s,j}+x_{t,1}))] \Delta^2 F(I_{s,j}) \nonumber \\ &\ = \exp(- i m x_{t,1}) \lim \frac{1}{M(s) h^2(s)} \sum_{j=- N(s)}^{N(s)} \exp(i m x_{s,j}) \Delta^2 F(I_{s,j}) \nonumber \\ &\ \qquad +\exp(i m x_{t,1}) \lim \frac{1}{M(s) h^2(s)} \sum_{j=- N(s)}^{N(s)} \exp(-i m x_{s,j}) \Delta^2 F(I_{s,j}). \end{aligned}
\end{equation}
\tag{5.17}
$$
Write (5.1) as follows:
$$
\begin{equation}
\begin{aligned} \, c_m+c_{-m} &= \lim \frac{1}{M(s) h^2(s)} \sum_{j=- N(s)}^{N(s)} \exp(i m x_{s,j}) \Delta^2 F(I_{s,j}) \nonumber \\ &\qquad +\lim\frac{1}{M(s) h^2(s)}\sum_{j=- N(s)}^{N(s)}\exp(-m x_{s,j}) \Delta^2 F(I_{s,j}). \end{aligned}
\end{equation}
\tag{5.18}
$$
Combining (5.17) with (5.18), we get the following system of two linear equations with variables $c_m$ and $c_{-m}$:
$$
\begin{equation*}
\begin{gathered} \, \begin{aligned} \, c_m+c_{-m} &= A_{-}+A_{+}, \\ \exp(i m x_{t,1})c_m+\exp(- i m x_{t,1})c_{-m} &= \exp(i m x_{t,1}) A_{-}+\exp(- i m x_{t,1})A_{+}, \end{aligned} \\ A_{\pm} := \lim \frac{1}{M(s) h^2(s)} \sum_{j=- N(s)}^{N(s)} \exp(\pm i m x_{s,j}) \Delta^2 F(I_{s,j}). \end{gathered}
\end{equation*}
\notag
$$
The determinant of the matrix of this system is $\exp(- i m x_{t,1})-\exp(i m x_{t,1})$; it does not vanish because
$$
\begin{equation*}
0 < m x_{t,1} = m \, \frac{2 \pi}{M(t)} \leqslant N(t) \, \frac{2 \pi}{M(t)} < \pi.
\end{equation*}
\notag
$$
Therefore, the system has a unique solution that is easy to see: $c_m\,{=}\,A_-$, $c_{-m}\,{=}\, A_+$. The above formulas are equivalent to (5.14). The Theorem is proved. Theorem 6. Suppose we are given a bijection $B \in \mathcal{B}_{\mathcal{M}}$, a trigonometric series $TS$ of the form (2.1), and a finite function $f$ defined and integrable on the set $G^q$ for some $q$. Assume that $(\mathrm{A}')$ the $B$-rearrangement of the series $TS$ converges to $f$ on the set $G^q \setminus H$; $(\mathrm{B}')$ the $\widehat{B}$-rearrangement of $TS$ converges to $f$ on $H$. Then for each $m \in \mathbb{Z}$ we have the equality
$$
\begin{equation}
c_m = \lim \frac{1}{M(s) h^2(s)} \sum_{j=- N(s)}^{N(s)} \exp(-i m x_{s,j}) \int_{I_{s,j}} f(u) (h(s)-|u-x_{s,j}|)\, du,
\end{equation}
\tag{5.19}
$$
where the limit is taken over all $s \in \mathbb{N}$ satisfying (5.2). Proof. First, note that the set $G^q \setminus H$ has a positive measure. Hence $(\mathrm{A}')$ and Proposition 2 imply that the series $TS$ satisfies (2.21). From this fact and $(\mathrm{B}')$ it follows that we are in the conditions of Theorems 4 and 5. Consequently, the formula (5.14) is true. Therefore, our theorem will be proved if, for each pair $s$ and $j$, $s \geqslant q$, we establish that $\Delta^2 F(I_{s,j})$ coincides with the integral in (5.19) (here $F$ is the Riemann function of the series $TS$). We will establish this fact.
Fix $s \geqslant q$ and $j$. From conditions $(\mathrm{A}')$ and $(\mathrm{B}')$ as well as from Theorem 2, it follows that the relation (2.10) with $A= f(x_0)$ holds for every $x_0 \in G^q$. Combining this fact with the continuity of the function $F(x)$, we obtain the equality
$$
\begin{equation}
F(x)=\int_{x_{s,j}-h(s)}^x dt \int_{x_{s,j}-h(s)}^t f(u) \, du +A x+B, \qquad x \in I_{s,j},
\end{equation}
\tag{5.20}
$$
for some constants $A$ and $B$, according to Theorem A1. For simplicity, we will write
$$
\begin{equation*}
a:= x_{s,j}-h(s),\qquad b := x_{s,j}, \qquad c := x_{s,j}+h(s).
\end{equation*}
\notag
$$
From (5.20) we get
$$
\begin{equation*}
\begin{aligned} \, \Delta^2 F(I_{s,j}) &= \int_a^c dt \int_a^t f(u) \, du - 2\int_a^b dt \int_a^t f(u) \, du + 0 \\ &= \int_b^c dt \int_a^t f(u) \, du - \int_a^b dt \int_a^t f(u) \, du \\ &= \int_b^c dt \biggl( \int_a^b f(u) \, du + \int_b^t f(u) \, du \biggr) - \int_a^b dt \int_a^t f(u) \, du \\ &= \int_b^c dt \int_b^t f(u) \, du + \int_a^b dt \int_a^b f(u) \, du - \int_a^b dt \int_a^t f(u) \, du \\ &= \int_b^c dt \int_b^t f(u) \, du + \int_a^b dt \int_t^b f(u) \, du \\ &= \int_b^c f(u) \, du \int_u^c dt + \int_a^b f(u) \, du \int_a^u dt \\ &= \int_b^c f(u)(c-u) \, du + \int_a^b f(u) (u-a) \, du \\ &= \int_a^c f(u) \biggl(\frac{c-a}{2} - |u-b| \biggr)\, du. \end{aligned}
\end{equation*}
\notag
$$
The expression on the right-hand side coincides with the integral (5.19), since $(c-a)/2=h(s)$, $b=x_{s,j}$. In our computations, we applied Fubini’s theorem, which holds because the two-variable function $g(t, u) := f(u)$ is absolutely integrable on the square $I_{s,j}^2$. The Theorem is proved. Theorem 7. Let $B \in \mathcal{B}_{\mathcal{M}}$ and $TS$ be a trigonometric series of the form (2.1). Assume that: $(\mathrm{A}'')$ the $B$-rearrangement of $TS$ converges to zero on the set $G^q \setminus H$ for at least one $q \in \mathbb{N}$; $(\mathrm{B}'')$ the $\widehat{B}$-rearrangement of $TS$ converges to zero on $H$. Then the coefficients of this series are all zero. Proof. Apply Theorem 6, taking $f \equiv 0$. According to formula (5.19) all $c_m=0$. This concludes the proof. Let us take as $B$ the identity map $\mathrm{Id} \colon \mathbb{Z} \to \mathbb{Z}$. Since $\mathrm{Id} \in \mathcal{B}_1 \subset \mathcal{B}_{\mathcal{M}}$, then from Theorem 7 we immediately get the following result that seems to be interesting in our opinion Theorem 8. If the series (2.1) converges in the usual sense to zero on the set $G^q \setminus H$ for at least one $q \in \mathbb{N}$, and its $\widehat{B}$-rearrangement converges to zero on $H$, then all coefficients of this series are zero. Theorem 9. If a series $\sum_{n \in \mathbb{Z}} c_{\widehat{B}(n)} \exp(i \widehat{B}(n) x)$ converges to zero on the set $G^q$ for some $q$, then the coefficients of the series $TS$ vanish. Proof. Applying again Theorem 7 by setting $B := \widehat{B}$, we obtain $\widehat{B} \in \mathcal{B}_{\mathcal{M}}$, by Lemma 5. Now it is clear that all the conditions of Theorem 7 hold. Consequently, all $c_m=0$. This completes the proof. Theorem 10 (Main Theorem). There exists a family $\mathcal{B}=\{B\}$ consisting of odd one-to-one mappings $B \colon \mathbb{Z} \to \mathbb{Z}$ with the following property. If $B \in \mathcal{B}$, then for the system $\{\exp(i B(n) x)\}_{n \in \mathbb{Z}}$ and each $\delta>0$ there are perfect $U$-sets of measure greater than $2 \pi -\delta$. Proof. We place in the class $\mathcal{B}$ the bijections $\widehat{B}=\widehat{B}(\mathbf{M}, \mathbf{h})$ taken for all possible sequences $\mathbf{M}=\{M(s)\}$ of odd numbers and $\mathbf{h}=\{h(s)\}$ of positive numbers, satisfying (4.1)–(4.5). For any bijection $\widehat{B}=\widehat{B}(\mathbf{M}, \mathbf{h})$ take the set $G^q=G^q(\mathbf{M}, \mathbf{h})$. It follows from Theorem 9 that the set $\mathbb{T} \setminus G^q$ is a $U$-set for the system $\{\exp(i \widehat{B}(n) x)\}_{n \in \mathbb{Z}}$ whenever $q \in \mathbb{N}$. Note that the sets $\mathbb{T} \setminus G^q$ are perfect (see (4.8)). It remains to apply this observation and Lemma 3. The Theorem is proved.
§ 6. DiscussionFirst, we note that the group properties of subsets $\mathbb{T}$ and $\mathbb{Z}$ are played an important role in constructing the class $\mathcal{B}$ of rearrangements of the trigonometric system and the corresponding $U$-sets of positive measure. Thus, the sets $G^q$ (of the Rajchman–Zygmund type), which are complements to $U$-sets, have a certain symmetry inside the group $\mathbb{T}$. Namely, they consist of intervals whose centers form a countable subgroup $\mathbb{T}$; moreover, $G^q$ is the union of a countable number of sets, each of which is invariant under a shift by a finite subgroup $\mathbb{T}$. The last properties are essential for the proof of the main results. When selecting rearrangements for class $\mathcal{B}$ their arithmetic structure was taken into account. Already it is clear from (2.14) and (2.15) that in the question of recovering the coefficients of a trigonometric series from its Riemann function, it is necessary to study the behavior of coefficients with large numbers forming a finite arithmetic progression. Note an interesting problem connected with Theorem 6, which deals with the possibility of recovering the coefficients of the series $BTS$ in the system $\{\exp(i B(n) x)\}_{n \in \mathbb{Z}}$ $(B \in \mathcal{B})$ in the case when it converges to a finite function $f \in L(\mathbb{T})$ on a special set $A$ of arbitrarily small measure. The question arises, whether the set $\mathbb{T} \setminus A$ is not $V$-set for the above system? To start answering this question, we note that for the case of sets of positive measure, the classical formulation of the concept of $V$-set loses its meaning. This is due to the fact that there are many ways to extend a function from a set of incomplete measure to the entire one-dimensional torus $\mathbb{T}$. In this connection, the coefficients of the series $BTS$ recovered by (5.19) do not have to be the Fourier coefficients of the function $f$. The following question immediately arises: under the conditions of Theorem 6, does the restriction of the function $f$ to the set $A$ always extend to the function $\widetilde{f}$ with domain $\mathbb{T}$ so that the series $BTS$ is the Fourier series of the function $\widetilde{f}$? And, if so, how? Let us say a few words about the relationship between the convergence of series in the trigonometric system and in its rearrangements. From the well-known papers by Carleson, Hunt, Sjölin and Antonov [30]–[33] it follows that the Fourier series of a function $f$ converges to it almost everywhere if $f \in L_p(\mathbb{T})$, $p>1$, and even for a number of logarithmic classes lying between $L_1(\mathbb{T})$ and all $L_p(\mathbb{T})$, $p>1$. At the same time, even earlier, Ul’yanov showed in [34] that trigonometric Fourier series, if rearranged, can diverge almost everywhere if $f \in L_p(\mathbb{T})$, $1 \leqslant p<2$. Thus, the rearranged Fourier series may behave worse than the original series, in terms of convergence. At the same time, there are no known examples of rearrangements for which Fourier series begin to behave better from the point of view of their convergence. The theory of uniqueness is in some sense dual to the theory of convergence. Therefore, it is natural to expect that, from the point of view of uniqueness, the rearranged trigonometric Fourier series may behave better than the original ones, in particular, that new $U$-sets may appear under certain rearrangements. The results of our work warrant these expectations. We also note the paper [35], in which it is proved that if for a given multiple trigonometric series any of its rearrangements converges to zero everywhere over cubes, then this series is trivial. If the original series is not rearranged, then it is not known whether there is uniqueness under convergence over cubes (see about this at the end of the paper). The results obtained in the paper are closely related to the following two open problems of the theory of uniqueness of trigonometric series. We have shown in Theorem 10 that there exist rearrangements of a trigonometric system for which there are $U$–sets of positive measure. And is it possible to state that at least $\varnothing$ is a $U$-set, but for any rearranged trigonometric system? This problem was already noted by Stechkin and Ul’yanov in [9]. The paper by Ash and Wang [26] contains a conjecture called the Stechkin–Ul’yanov conjecture that the answer to this question is yes. According to Theorem 3, the conjecture is confirmed for $D$-monotone rearrangements, even in a stronger form, since instead of $U$-sets we can talk about $V$-sets. Another open problem has been repeatedly stated in a number of works (see, for example, [36], [37]) and is formulated as follows. Is it true that among sets of positive measure there are no $U$-sets for multidimensional trigonometric series? The answer to the last question may depend on the type of convergence. In our work, the following consideration leads to this problem. If in the one-dimensional case it was possible to construct $U$-sets of positive measure for rearrangements of the trigonometric system, then perhaps in the multidimensional case such sets can already be constructed for the natural summation order, if we consider one of the strong types of convergence. So far, the following is known about $U$–sets for the multidimensional trigonometric system. Tetunashvili constructed [38] wide classes of continual $U$- and $V$-sets for Pringsheim (rectangular) convergence. These classes have been extended in [16], [39] and [40]; they all consist of sets of zero measure. And for convergence over cubes, it is still unknown even whether at least the empty set is a $U$-set. Ash conjectured that this is not true. For spherical convergence, Bourgain proved [41] that $\varnothing$ is a $U$-set. 
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