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Density of the sums of shifts of a single function in the N. A. Dyuzhinaab a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia b Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia
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     § 1. IntroductionIn the real space $L^{0}_{p}(\mathbb{T})$ of mean zero functions on the circle $\mathbb{T}$ that are integrable to power $p$, $1\leqslant p <\infty$, there exists a function $f$ such that the sums of shifts of $f$ are dense in $L^{0}_{p}(\mathbb{T})$ (see [1], where whole classes of such functions were distinguished). There exists a function on the real line $\mathbb{R}$ such that the sums of shifts of it are dense in all real spaces $L_{p}(\mathbb{R})$ for $2 \leqslant p <\infty$ [2]. The real space $l_{2}(\mathbb{Z})$ of two-sided sequences contains an element such that the sums of its shifts are dense in all real spaces $l_{p}(\mathbb{Z})$ for $2 \leqslant p <\infty$ (see [3]). In [4] these results were extended to several dimensions, that is, to the torus $\mathbb{T}^{d}$, the space $\mathbb{R}^{d}$ and the lattice $\mathbb{Z}^{d}$, where $d \in \mathbb{N}$. In [5] sufficient conditions on a function $f$ were found under which the sums of functions of the form $f(\alpha x - \theta)$, $ \alpha \in A \subset \mathbb{R}$, $\theta \in \Theta \subset \mathbb{R}$, are dense in the real space $L_{p}$ on the line or a compact subset of it. For a survey of results on the density of the sums of shifts of a single function, see [6]. In this connection the following problem arises in the natural way, which was stated in [6]. Problem 1. Let $G$ be a locally compact Abelian group with Haar measure $m$. Does there exist a function $f$ on this group such that the sums of shifts of $f$: The compact case is distinguished in Problem 1 for the following reason. If $G$ is a compact Abelian group, then its Haar measure satisfies $m(G) < \infty$ (see [7], Ch. 4, § 15, Theorem 15.9). Then for a function $f \in L_{2}(G)$ the sums (1.1) of shifts of $f$ cannot be dense in the whole of $L_{2}(G)$: the mean value $\displaystyle \int_{G}f(g)\,dm(g)=\alpha$ of $f$ is defined and sums (1.1) cannot approximate functions with mean value outside the set $\{n\alpha\colon n \in \mathbb{N} \}$. The aim of this paper is to give an exhaustive answer to question (b) in Problem 1, namely, establish the following result. Theorem 1. Let $G$ be a nontrivial compact Abelian group. Then a function $f$: $G \to \mathbb{R}$ such that the sums (1.1) of shifts of $f$ are dense in the real space $L_{2}^{0}(G)$ exists if and only if $G$ is connected and its character group is infinite countable. Note that if $G$ is a trivial group consisting of the single neutral element, then the space $L_{2}^{0}(G)$ contains only the function identically equal to zero, the sums of shifts of which are dense in this space. § 2. Auxiliary lemmasThroughout what follows $f_{g}(\,\cdot\,)=f(\,\cdot+g)$ is the shift of $f$ by the element $g$ of $G$. We prove a few auxiliary lemmas. Lemma 1. If $G$ is a finite nontrivial Abelian group, then there exists no function $f\colon G \to \mathbb{R}$ such that the sums (1.1) of shifts of it are dense in the real space $L_{2}^{0}(G)$. Proof. We fix a real function $f \in L_{2}^{0}(G)$ and number the elements of the group: $G= \{g_{1}, g_{2}, \dots , g_{N} \}$, $ N \geqslant 2$. Each $g_{k}$ is an atom of the measure $m$, and $m(g_{k})=1/N$. We represent a function $h\colon G \to \mathbb{R}$ as the vector of values $(h(g_{1}), h(g_{2}), \dots , h(g_{N}))$, and we represent $L_{2}^{0}(G)$ as the space $L$ of vectors of length $N$ with sum of coordinates equal to zero. Since $f$ has mean value zero, for each $g \in G$ we have $\sum_{n=1}^{N} f(g+g_{n})=0$. Therefore,
Consider the set of sums of shifts of $f$:
$$
\begin{equation}
S :=\biggl\{\sum_{k=1}^{m} f_{h_{k}}\colon h_{k} \in G, \ m \in \mathbb{N} \biggr\} =\biggl\{\sum_{n=1}^{N} \nu_{n}f_{g_{n}}\colon \nu_{n} \in \mathbb{N} \cup \{0\} \biggr\}.
\end{equation}
\tag{2.2}
$$
From (2.1) and (2.2) we obtain
$$
\begin{equation*}
\begin{aligned} \, S&=\biggl\{\sum_{n=1}^{N-1} \nu_{n}f_{g_{n}} - \nu_{N}\biggl( \sum_{n=1}^{N-1} f_{g_{n}} \biggr)\colon \nu_{n} \in \mathbb{N} \cup \{0\} \biggr\} \\ &=\biggl\{\sum_{n=1}^{N-1} (\nu_{n} - \nu_{N})f_{g_{n}}\colon \nu_{n} \in \mathbb{N} \cup \{0\} \biggr\} =\biggl\{\sum_{n=1}^{N-1} \lambda_{n}f_{g_{n}}\colon \lambda_{n} \in \mathbb{Z} \biggr\}. \end{aligned}
\end{equation*}
\notag
$$
If the vectors $f_{g_{n}}$, $ n=1,\dots ,N-1$, are linearly independent, then the set $S$, and therefore also its closure $\overline{S}$, is the integer lattice generated by these vectors, so the closure $\overline{S}$ cannot coincide with the $(N-1)$-dimensional space $L$. If the vectors $f_{g_{n}}$, $ n=1,\dots ,N-1$, are linearly dependent, then $\overline{S}$ lies in a subspace of dimension at most $N-2$, so it cannot coincide with $L$ of dimension $N-1$.
Lemma 1 is proved. Lemma 2. Let $G$ be a compact Abelian group and $H$ be a closed subgroup of $G$. If there exists a function $f_{0}\colon G \to \mathbb{R}$ the sums of shifts of which are dense in the real space $L_{2}^{0}(G)$, then there exists a function $F_{0}\colon G/H \to \mathbb{R}$ the sums of shifts of which are dense in the real space $L_{2}^{0}(G/H)$. Proof. Since $G$ is a compact Abelian group and $H$ is a closed subgroup of it, $G/H$ is a compact Abelian group (see [7], Ch. 2, § 5, Theorem 5.22, and [8], Appendix B, § B6). We denote the coset of an element $x \in G$ by the subgroup $H$ by $\widehat{x}$. According to [8], Ch. 2, § 2.7.3, the groups $G, H$ and $G/H$ are endowed with the Haar measures $m_{G}$, $m_{H}$ and $ m_{G/H}$ such that $m_{H}(H)=1$ and for each function $f \in L_{1}(G)$ the function is well defined on $G/H$; moreover, the map $T\colon f \mapsto F$ is a bounded linear operator $T\colon L_{1}(G) \to L_{1}(G/H)$ and
$$
\begin{equation}
\int_{G}f\,dm_{G}=\int_{G/H}F(\widehat{x})\,dm_{G/H}(\widehat{x}).
\end{equation}
\tag{2.4}
$$
Since $f_{0} \in L_{2}^{0}(G)$ and $G$ is a compact group, it follows that $f_{0} \in L_{1}(G)$ and the function $F_{0}=Tf_{0} \in L_{1}(G/H)$ is well defined. Using equality (2.4) for the functions $f_{0}$, $|f_{0}|^{2} \in L_{1}(G)$, we obtain
$$
\begin{equation*}
\begin{aligned} \, &\int_{G/H}F_{0}(\widehat{x})\,dm_{G/H}(\widehat{x})=\int_{G}f_{0}\,dm_{G}=0, \\ &\int_{G/H}|F_{0}(\widehat{x})|^{2}\,dm_{G/H}(\widehat{x})=\int_{G/H} \biggl| \int_{H}f_{0}(x+y)\,dm_{H}(y) \biggr|^{2}\,dm_{G/H}(\widehat{x}) \\ &\qquad \leqslant \int_{G/H} \int_{H} |f_{0}(x+y)|^{2}\,dm_{H}(y)\,dm_{G/H}(\widehat{x}) \\ &\qquad= \int_{G}|f_{0}(x)|^{2}\,dm_{G}(x)=\| f_{0} \|_{L_{2}(G)}^{2} < \infty, \end{aligned}
\end{equation*}
\notag
$$
that is, $F_{0} \in L_{2}^{0}(G/H)$.
Now we show that the sums of shifts of $F_{0}$ are dense in the space $L_{2}^{0}(G/H)$. We fix $P \in L_{2}^{0}(G/H)$ and define a function $p$ on $G$ by $p(g):=P(\widehat{g})$. Clearly, $P=Tp$. By Theorem 3 in [9], Ch. VIII, § 39, $p$ is a function in $L_{1}(G)$, and we have
$$
\begin{equation*}
\int_{G}p\,dm_{G}=\int_{G/H} P\,dm_{G/H}=0, \qquad \int_{G}|p|^{2}\,dm_{G}= \int_{G/H} |P|^{2}\,dm_{G/H} < \infty .
\end{equation*}
\notag
$$
Thus, $p \in L_{2}^{0}(G)$, and for each $\varepsilon>0$ there exist by assumption $n \in \mathbb{N}$ and $ g_{k} \in G$, $ k=1,\dots ,n$, such that
$$
\begin{equation}
\biggl\| p(g) - \sum_{k=1}^{n}f_{0}(g+g_{k}) \biggr\|_{L_{2}(G)} < \varepsilon.
\end{equation}
\tag{2.5}
$$
Using the definition of $F_{0}$, equalities (2.3) and (2.4) for the function $|p(\,\cdot\,) - \sum_{k=1}^{n}f_{0}(\,\cdot+g_{k})|^{2}$ and inequality (2.5) we obtain
$$
\begin{equation*}
\begin{aligned} \, &\biggl\| P(\widehat{g}) - \sum_{k=1}^{n}F_{0}(\widehat{g}+\widehat{g_{k}}) \biggr\|_{L_{2}(G/H)}^{2} \\ &\qquad =\int_{G/H} \biggl| \int_{H} \biggl( p(g+y) - \sum_{k=1}^{n}f_{0}(g+g_{k}+y) \biggr)\,dm_{H}(y) \biggr|^{2}\,dm_{G/H}(\widehat{g}) \\ &\qquad \leqslant \int_{G/H} \biggl( \int_{H} \biggl| p(g+y) - \sum_{k=1}^{n}f_{0}(g+g_{k}+y) \biggr|^{2}\,dm_{H}(y) \biggr)\,dm_{G/H}(\widehat{g}) \\ &\qquad =\int_{G} \biggl| p(g) - \sum_{k=1}^{n}f_{0}(g+g_{k}) \biggr|^{2}\,dm_{G}(g) < \varepsilon^{2}. \end{aligned}
\end{equation*}
\notag
$$
This means exactly that the sums of shifts of $F_{0}$ are dense in $L_{2}^{0}(G/H)$.
Lemma 2 is proved. Let $G^{*}$ denote the group of continuous characters of $G$, and $\mathbb{I}$ denote the unit character on $G$. Let $H$ be a closed subgroup of the locally compact Abelian group $G$ and $H^{\perp}$ be the set of $\gamma \in G^{*}$ such that $\gamma(h)=1$ for all $h \in H$. Then $H^{\perp}$ is called the annihilator of $H$. By [8], Ch. 2, § 2.1.1, $H^{\perp}$ is a closed subgroup of $G^{*}$. Lemma 3. Let $G$ be a disconnected compact Abelian group. Then there does not exist a function $f$ in the real space $L_{2}^{0}(G)$ such that the sums of shifts (1.1) of $f$ are dense in this space. Proof. By [7], Ch. 6, § 24, Theorem 24.25, the character group $G^{*}$ of a disconnected compact Abelian group $G$ has torsion: it contains a nontrivial element $\chi_{0} \in G^{*}$ of finite order $n_{0} \geqslant 2$. Therefore, $\Xi :=\{\chi_{0}, \chi_{0}^{2}, \dots , \chi_{0}^{n_{0}} \equiv \mathbb{I} \} $ is a closed subgroup of $G^{*}$ of order $n_{0}$. Let $H=\Xi^{\perp}$ be the annihilator of $\Xi \subset G^{*}$. Then $H$ is a closed subgroup of $(G^{*})^{*}$, so that by Pontryagin’s duality theorem (see [8], Ch. 1, § 1.7.2) $H$ is a closed subgroup of $G$ and the quotient group $G/H$ coincides with $(G^{*})^{*}/\,\Xi^{\perp}$. By [8], Ch. 2, § 2.1.2, the quotient group $(G^{*})^{*}/\,\Xi^{\perp}$ is topologically isomorphic to $\Xi^{*}$. By [7], Ch. 6, § 23.27.d, the character group $\Xi^{*}$ of the finite Abelian group $\Xi$ is topologically isomorphic to $\Xi$. Thus, the disconnected compact Abelian group $G$ contains a closed subgroup $H$ such that $G/H$ is topologically isomorphic to a finite group $\Xi$ of order $n_{0} \geqslant 2$.
Assume that there exists a function $f$ in the real space $L_{2}^{0}(G)$ such that the sums (1.1) are dense in this space. Then by Lemma 2 there exists a function $F$ in the real space $L_{2}^{0}(G/H)$ such that the sums of shifts of $F$ are dense in this space. However, $G/H$ is a nontrivial finite Abelian group. This is in contradiction to Lemma 1. Lemma 3 is proved. Lemma 4. Let $G$ be a nontrivial compact Abelian group such that its character group $G^{*}$ is not infinite countable. Then in the real space $L_{2}^{0}(G)$ there exists no function $f$ such that the sums of shifts (1.1) are dense in this space. Proof. If $G^{*}$ is finite, then by [7], Ch. 6, § 23.27.d, $G$ is topologically isomorphic to $G^{*}$, so $G$ is a nontrivial finite Abelian group and the required result follows from Lemma 1.
Consider the case when $G^{*}$ is uncountable. Let $f \in L_{2}^{0}(G)$. Then by the completeness of the system of characters of a compact Abelian group ([10], Ch. III, § 2, Theorem 3.9) the character group $G^{*}$ is an orthonormal basis of $L_{2}(G)$; in particular, $f$ expands in a Fourier series in the system of characters:
$$
\begin{equation*}
f(g)=\sum_{\alpha}c_{\alpha}\chi_{\alpha}(g), \qquad G^{*}=\{\chi_{\alpha} \},
\end{equation*}
\notag
$$
where the set of nonzero coefficients $c_{\alpha}$ is at most countable (otherwise Parseval’s identity does not hold). Hence
$$
\begin{equation*}
f(g)=\sum_{k \in \mathbb{N}}c_{\alpha_{k}}\chi_{\alpha_{k}}(g), \qquad c_{\alpha_{k}} \neq 0,
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
f(g+h)=\sum_{k \in \mathbb{N}}c_{\alpha_{k}}\chi_{\alpha_{k}}(h)\chi_{\alpha_{k}}(g).
\end{equation*}
\notag
$$
Then sums of shifts of $f$ lie in the closed subspace $L$ of the real space $L_{2}^{0}(G)$ that is spanned by the functions $\chi_{\alpha_{k}}$, $ k \in \mathbb{N}$, and, as a basis of $L_{2}^{0}(G)$ is uncountable, $L$ does not coincide with $L_{2}^{0}(G)$.
Lemma 4 is proved. § 3. Proof of Theorem 1 Proof. Necessity. This follows from Lemmas 3 and 4.
Sufficiency. Now let $G$ be a connected compact Abelian group with infinite countable character group $G^{*}$. We prove in several steps that the required function exists. 1. Let $\chi$ be a continuous character on the group $G$, $\mathbf{0}$ be the identity element of $G$, and $\mathbb{I}$ be the neutral element of $G^{*}$. Then $\overline{\chi}$ is also a continuous character on $G$, and $\chi \equiv \overline{\chi}$ on $G$ if and only if $\chi$ takes only the values $\pm 1$ in $G$. However, $\chi(\mathbf{0})=1$ and $G$ is a connected group, so $\chi \equiv \overline{\chi}$ if and only if $\chi \equiv \mathbb{I}$. Thus, the group $G^{*}$ has the form
$$
\begin{equation*}
G^{*}=\{\chi_{\nu} \}_{\nu=1}^{\infty} \sqcup \{\overline{\chi}_{\nu} \}_{\nu=1}^{\infty} \sqcup \{\mathbb{I} \}.
\end{equation*}
\notag
$$
Note that $G^{*}$ has a discrete topology (see [8], Ch. 1, § 2, Theorem 1.2.5), and compact subsets of $G^{*}$ are merely finite subsets of it. By the completeness of the system of characters of a compact Abelian group (see [10], Ch. III, § 2, Theorem 3.9) $G^{*}$ forms an orthonormal basis of $L_{2}(G)$, and each real function $f \in L_{2}^{0}(G)$ expands in a Fourier series in the system $G^{*}$:
$$
\begin{equation*}
f(g)=\sum_{\nu=1}^{\infty}c_{\nu}\chi_{\nu}(g)+\sum_{\nu=1}^{\infty}\overline{c_{\nu}} \overline{\chi}_{\nu}(g), \qquad c_{\nu} \in \mathbb{C}.
\end{equation*}
\notag
$$
We seek a function $f$ such that the sums of shifts of $f$ are dense in $L_{2}^{0}(G)$ in the following form:
$$
\begin{equation}
f(g)=\sum_{\nu=1}^{\infty}c_{\nu}(\chi_{\nu}(g)+\overline{\chi}_{\nu}(g)), \qquad c_{\nu} \in \mathbb{R}.
\end{equation}
\tag{3.1}
$$
2. By [7], Ch. 6, § 24, Theorem 24.15, the topological weight $\mu(G)$ of the compact Abelian group $G$ coincides with the cardinality of $G^{*}$, that is, it is infinite countable. By [7], Ch. 6, § 25, Theorem 25.14, if the topological weight $\mu(G)$ of a connected compact Abelian group $G$ does not exceed the cardinality of a continuum, then $G$ is monothetic, that is, there exists $g_{0} \in G$ such that $\overline{\{ng_{0}\colon n \in \mathbb{Z} \}}=G$. By [7], Ch. 6, § 25, Theorem 25.11, each nontrivial character is distinct from 1 at the element $g_{0}$: $\chi_{\nu}(g_{0}) \neq 1$, $ \nu \in \mathbb{N}$. Therefore,
$$
\begin{equation}
\forall\, k \in \mathbb{N} \quad \exists\, \delta_{k}>0\colon \quad|\chi_{\nu}(g_{0}) - 1| \geqslant \delta_{k} \quad \text{for } \nu=1,\dots,k.
\end{equation}
\tag{3.2}
$$
For each $k \in \mathbb{N}$ fix $\varepsilon_{k} \in (0, \delta_{k}/k)$. By Dirichlet’s theorem on simultaneous approximation (see [11], Ch. 1, § 5) there exists a sequence of positive integers $\{N_{k} \}_{k=0}^{\infty}$ such that
$$
\begin{equation}
N_{0}=1, \qquad N_{k} \geqslant kN_{k-1} \quad \text{and} \quad |(\chi_{\nu}(g_{0}))^{N_{k}} - 1| < \varepsilon_{k}, \qquad \nu=1,\dots,k, \quad k \in \mathbb{N};
\end{equation}
\tag{3.3}
$$
in particular, the sequence $\{N_{k} \}_{k=0}^{\infty}$ satisfies the condition
$$
\begin{equation}
N_{k+m} \geqslant (k+m)(k+m-1)\dotsb(k+1)N_{k}, \qquad k, m \in \mathbb{N}.
\end{equation}
\tag{3.4}
$$
3. By [7], Ch 6, § 25, Theorem 25.18, 2 $\Rightarrow$ 3, given a connected compact Abelian group $G$, there exists a homomorphism $\varphi\colon G^{*} \to \mathbb{R}_{d}$ into the additive group $\mathbb{R}_{d}$ of real numbers endowed with the discrete topology. According to another part of the same result (see [7], Ch. 6, § 25, Theorem 25.18, 3 $\Rightarrow$ 1), $G$ is solenoidal, that is, there exists a continuous homomorphism $\tau \colon \mathbb{R} \to G$ such that $\overline{\tau(\mathbb{R})}=G$, and we can see from the proof that
$$
\begin{equation}
\forall\, \chi \in G^{*}, \quad \forall\, t \in \mathbb{R}\colon\quad \chi(\tau(t))= \exp(it\varphi(\chi)).
\end{equation}
\tag{3.5}
$$
Set
$$
\begin{equation}
a_{m} :=\min \biggl\{\frac{1}{2^{m}}, \frac{1}{(m^{2}|\varphi(\chi_{m})|)}\biggr \}, \qquad m \in \mathbb{N}.
\end{equation}
\tag{3.6}
$$
Here $\varphi(\chi_{m}) \ne 0$ because otherwise $\chi_{m} \equiv 1$ by identity (3.5) since the image of $\tau$ is dense in $G$.
4. We show that the function
$$
\begin{equation}
\rho(g, h) :=\sum_{m=1}^{\infty}a_{m} |\chi_{m}(g) - \chi_{m}(h)|, \qquad g, h \in G,
\end{equation}
\tag{3.7}
$$
is a metric on $G$. The function $\rho$ is well defined because equalities (3.6) imply the estimate $\rho(g,h) \leqslant \sum_{m=1}^{\infty} 1/2^{m-1}=2$ for $ g, h \in G$. Clearly, $\rho$ is nonnegative, symmetric and, by the triangle inequality for the modulus, satisfies the triangle inequality. If $\rho(g, h)=0$, then $\chi_{m}(g)=\chi_{m}(h)$ for each $m \in \mathbb{N}$, so that $(\alpha(g- h))(\chi)=\chi(g-h)=1$ for all $\chi \in G^{*}$, where $ \alpha\colon G \to (G^{*})^{*}$ is the canonical isomorphism ([8], Ch. 1, §§ 1.7.1–1.7.2). Therefore, $\alpha(g-h)$ is the identity element of the group $(G^{*})^{*}$, and so $g-h= \mathbf{0}$. It follows from (3.7) that
$$
\begin{equation}
|\chi_{m}(g) - \chi_{m}(h)|=|\overline{\chi}_{m}(g) - \overline{\chi}_{m}(h)| \leqslant \frac{\rho(g,h)}{a_{m}}, \qquad m \in \mathbb{N},
\end{equation}
\tag{3.8}
$$
so that each character $\chi \in G^{*}$ is a Lipschitz function with respect to $\rho$. Moreover, it is obvious from the definition of $\rho$ that this metric is shift invariant.
Now we show that the topology on $G$ induced by $\rho$ coincides with the topology of the group $G$. By [8], Ch. 1, § 1.2.6, and [8], Ch. 1, § 1.7.2, a basis of topology on $G$ consists of the sets
$$
\begin{equation*}
N(x, C, r)=\{y \in G\colon |\gamma(y)-\gamma(x)|<r\text{ for all }\gamma \in C \},
\end{equation*}
\notag
$$
where $x \in G$, $C$ is a compact subset of $G^{*}$ and $r>0$. First we show that for all $\varepsilon>0$ and $x \in G$ there exist a compact set $C \subset G^{*}$ and $r>0$ such that $N(x, C, r) \subset B_{\varepsilon}(x) :=\{y \in G\colon\rho(x, y) < \varepsilon \}$. We choose $M \in \mathbb{N}$ such that ${1/2^{M} < \varepsilon/4}$, and set $r:= \varepsilon/2$ and $C:=\{\chi_{1},\dots ,\chi_{M} \}$. Then for all $y \in N(x, C, r)$ and $m=1,\dots,M$ we have the inequality $|\chi_{m}(y)-\chi_{m}(x)|<r$, so that by the definition of the coefficients $a_{m}$, $m \in \mathbb{N}$,
$$
\begin{equation*}
\rho(x, y) \leqslant \sum_{m=1}^{M} a_{m} |\chi_{m}(y) - \chi_{m}(x)|+\sum_{m=M+1}^{\infty} \frac{1}{2^{m-1}} < \sum_{m=1}^{M} \frac{r}{2^{m}}+\frac{1}{2^{M-1}} < \varepsilon,
\end{equation*}
\notag
$$
that is, $y \in B_{\varepsilon}(x)$.
Next we show that for all $r>0$ and $ x \in G$ and each compact set $C \subset G^{*}$ there exists $\varepsilon>0$ such that $B_{\varepsilon}(x) \subset N(x, C, r)$. Because $G^{*}$ is infinite countable and $C \subset G^{*}$ is a compact set, $C$ is finite and there exists $M \in \mathbb{N}$ such that $C \subset \{\mathbb{I}, \chi_{1}, \overline{\chi}_{1}, \dots , \chi_{M}, \overline{\chi}_{M} \}$. Set $\varepsilon:=r\min_{m=1, \dots , M}a_{m}$. If $y \in B_{\varepsilon}(x)$, then by (3.8) we have $|\chi_{m}(y) - \chi_{m}(x)|=|\overline{\chi}_{m}(y) - \overline{\chi}_{m}(x)| < \varepsilon/a_{m} \leqslant r$ for $m=1,\dots , M$, that is, $y \in N(x, C, r)$. We see that the metric $\rho$ agrees with the topology of the group $G$. 5. Now we prove that the continuous homomorphism $\tau\colon \mathbb{R} \to G$ defined in part 3 of the proof is Lipschitz with respect to the metric $\rho$ on $G$. Let $u, v \in \mathbb{R}$. Then from (3.5) and (3.7) we obtain
$$
\begin{equation*}
\begin{aligned} \, \rho(\tau(u), \tau(v)) &=\sum_{m=1}^{\infty}a_{m} |\exp(iu\varphi(\chi_{m})) -\exp(iv\varphi(\chi_{m}))| \\ &\leqslant |u-v| \biggl( \sum_{m=1}^{\infty} a_{m}|\varphi(\chi_{m})| \biggr) \leqslant |u-v| \biggl( \sum_{m=1}^{\infty} \frac{1}{m^{2}} \biggr) \leqslant 2|u-v|, \end{aligned}
\end{equation*}
\notag
$$
where in the penultimate inequality we used the definition (3.6) of the $a_{m}$.
6. For each $\nu \in \mathbb{N}$ we choose a constant $c_{\nu}$ so that
$$
\begin{equation}
0 < c_{\nu} < \min \biggl\{\frac{1}{N_{\nu}},\frac{a_{\nu}}{\nu} \biggr\},
\end{equation}
\tag{3.9}
$$
where $N_{\nu}$ and $a_{\nu}$ were defined in parts 2 and 3 of the proof, respectively. Then the function $f$ is defined by (3.1). Using inequalities (3.4) and (3.9) we can estimate the norm of $f$ in $L_{2}(G)$:
$$
\begin{equation*}
\|f\|_{2}^{2}= 2\sum_{\nu=1}^{\infty}|c_{\nu}|^{2} < 2\sum_{\nu=1}^{\infty}\frac{1}{N_{\nu}^{2}} \leqslant 2 \sum_{\nu=1}^{\infty} \frac{1}{N_{1}^{2}(\nu !)^{2}} \leqslant 4.
\end{equation*}
\notag
$$
Hence $f \in L_{2}^{0}(G)$. Next we estimate the following norm:
$$
\begin{equation*}
\begin{aligned} \, &\biggl\|\sum_{m=0}^{N_{k}-1} f(x+mg_{0}) \biggr\|_{2}^{2}=\biggl\| \sum_{m=0}^{N_{k}-1} \sum_{\nu=1}^{\infty} \bigl( c_{\nu} \chi_{\nu}(x) (\chi_{\nu}(g_{0}))^{m}+c_{\nu} \overline{\chi}_{\nu}(x) (\overline{\chi}_{\nu}(g_{0}))^{m} \bigr) \biggr\|_{2}^{2} \\ &\qquad =\biggl\| \sum_{\nu=1}^{\infty} c_{\nu} \biggl( \sum_{m=0}^{N_{k}-1} (\chi_{\nu}(g_{0}))^{m} \biggr) \chi_{\nu}(x)+\sum_{\nu=1}^{\infty} c_{\nu} \biggl( \sum_{m=0}^{N_{k}-1} (\overline{\chi}_{\nu}(g_{0}))^{m} \biggr) \overline{\chi}_{\nu}(x) \biggr\|_{2}^{2} \\ &\qquad =2 \sum_{\nu=1}^{\infty} \biggl| c_{\nu} \sum_{m=0}^{N_{k}-1} (\chi_{\nu}(g_{0}))^{m} \biggr|^{2} \leqslant 2 \sum_{\nu=1}^{k} |c_{\nu}|^{2} \biggl| \frac{(\chi_{\nu}(g_{0}))^{N_{k}} - 1}{\chi_{\nu}(g_{0}) - 1} \biggr|^{2}+2 \sum_{\nu=k+1}^{\infty}|c_{\nu}|^{2} N_{k}^{2}. \end{aligned}
\end{equation*}
\notag
$$
Using conditions (3.2)–(3.4) and (3.9) and the definition of the constants $\varepsilon_{k}$ we obtain
$$
\begin{equation*}
\begin{aligned} \, &\biggl\|\sum_{m=0}^{N_{k}-1} f(x+mg_{0}) \biggr\|_{2}^{2} \leqslant 2 \sum_{\nu=1}^{k} \frac{1}{N_{\nu}^{2}} \biggl( \frac{\varepsilon_{k}}{\delta_{k}} \biggr)^{2}+2 \sum_{\nu=k+1}^{\infty} \frac{N_{k}^{2}}{N_{\nu}^{2}} \\ &\qquad\leqslant \frac{2}{k^{2}} \sum_{\nu=1}^{k} \frac{1}{N_{\nu}^{2}}+2 \sum_{\nu=k+1}^{\infty} \frac{N_{k}^{2}}{\nu^{2} (\nu -1)^{2}\dotsb (k+1)^{2}N_{k}^{2}} \\ &\qquad \leqslant \frac{2}{k^{2}}k+\frac{2}{(k+1)^{2}}\biggl( 1+\frac{1}{2^{2}}+\frac{1}{3^{2}}+ \dotsb\biggr) \leqslant \frac{2}{k}+\frac{4}{(k+1)^{2}} \to 0, \qquad k \to \infty. \end{aligned}
\end{equation*}
\notag
$$
Hence $-f \in S$, where
$$
\begin{equation*}
S=\overline{\biggl\{\sum_{k=1}^{n} f(x+h_{k}),\,h_{k} \in G,\,n \in \mathbb{N} \biggr\}}
\end{equation*}
\notag
$$
(the closure in $L_{2}(G)$). Therefore, $-f(\,\cdot+h)$ belongs to $S$ for all $h \in G$, that is, $S$ is a closed additive subgroup of $L_{2}^{0}(G)$.
7. We require the following result. Lemma A ([12], Lemma 4). Let $S$ be a closed additive subgroup of a uniformly smooth Banach space $X$ with modulus of smoothness $s(t)$, $ t \geqslant 0$. If $a, b \in S$ and for each $\varepsilon > 0$ there exist $x_{0},\dots ,x_{n} \in S$ such that $x_{0}=a$, $ x_{n}=b$ and $\sum_{k=1}^{n}s(\|x_{k}-x_{k-1}\|) < \varepsilon$, then the whole line segment $[a, b]$ lies in $S$. Let $h \in G$ and $\varepsilon>0$. Since $\tau\colon \mathbb{R} \to G$ is a homomorphism with dense image and the metric $\rho$ agrees with the topology of $G$ (see parts 3 and 4 of the proof), there exists $w \in \mathbb{R}$ such that $\rho(\tau(w), h)< \sqrt{\varepsilon/2}$. Fix an integer $N > 1+8|w|^{2}/\varepsilon$ and set
$$
\begin{equation*}
h_{k}:=\tau \biggl( \frac{kw}{N-1} \biggr), \qquad k=0,\dots ,N-1, \quad h_{N}:=h.
\end{equation*}
\notag
$$
As the metric $\rho$ is translation invariant and $\tau$ is a bi-Lipschitz homomorphism (see parts 4 and 5 of the proof), we obtain the estimate
$$
\begin{equation*}
\begin{aligned} \, & \sum_{k=1}^{N}(\rho(h_{k-1}, h_{k}))^{2}=\sum_{k=1}^{N-1} \biggl( \rho \biggl( \tau \biggl( \frac{(k-1)w}{N-1} \biggr), \tau \biggl(\frac{kw}{N-1} \biggr) \biggr) \biggr)^{2}+ (\rho(\tau(w), h))^{2} \\ &\qquad < (N-1)\biggl( \rho \biggl( \tau(0), \tau \biggl( \frac{w}{N-1} \biggr) \biggr) \biggr)^2+ \frac{\varepsilon}{2} \leqslant (N-1)\cdot 4 \biggl|\frac{w}{N-1} \biggr|^{2}+ \frac{\varepsilon}{2} < \varepsilon. \end{aligned}
\end{equation*}
\notag
$$
We can estimate the sum
$$
\begin{equation*}
\sum_{k=1}^{N} \bigl\|f(x+h_{k}) - f(x+h_{k-1}) \bigr\|_{2}^{2}
\end{equation*}
\notag
$$
by using the above bound and inequalities (3.8) and (3.9):
$$
\begin{equation}
\begin{aligned} \, \notag &\sum_{k=1}^{N} \bigl\|f(x+h_{k}) - f(x+h_{k-1}) \bigr\|_{2}^{2} \\ \notag &\qquad =\sum_{k=1}^{N} \biggl\| \sum_{\nu=1}^{\infty} (c_{\nu}(\chi_{\nu}(h_{k}) - \chi_{\nu}(h_{k-1})) \chi_{\nu}(x)+c_{\nu}(\overline{\chi}_{\nu}(h_{k}) - \overline{\chi}_{\nu}(h_{k-1})) \overline{\chi}_{\nu}(x)) \biggr\|_{2}^{2} \\ \notag &\qquad =2 \sum_{k=1}^{N} \sum_{\nu=1}^{\infty} |c_{\nu}(\chi_{\nu}(h_{k}) - \chi_{\nu}(h_{k-1}))|^{2} \leqslant 2 \biggl( \sum_{\nu=1}^{\infty} \frac{c_{\nu}^{2}}{a_{\nu}^{2}} \biggr) \biggl( \sum_{k=1}^{N}(\rho(h_{k-1}, h_{k}))^{2} \biggr) \\ &\qquad < 2 \varepsilon \sum_{\nu=1}^{\infty} \frac{1}{\nu^{2}} < 4\varepsilon. \end{aligned}
\end{equation}
\tag{3.10}
$$
Thus, the subgroup $S$ defined in part 6 of the proof lies in the space $L_{2}^{0}(G)$ with modulus of smoothness $s(t)=\sqrt{1+t^{2}} - 1=O(t^{2})$ (for instance, see [13], Ch. 1, § e) and moreover, the functions $f(x+h_{0})=f(x)$, $ f(x+h_{1})$, $\dots $, $ f(x+h_{N})=f(x+h)$ belong to $S$ and (3.10) holds. Then by Lemma A, for each $\lambda \in [0,1]$ the function $\lambda f(x)+(1- \lambda) f(x+h)$ belongs to $ S$. Hence for each $\lambda \in \mathbb{R}$ we also have $\lambda (f(x)- f(x+h)) \in S$. Therefore, $S$ contains the closed $\mathbb{R}$-linear subspace $L$ spanned by the functions of the form $f(\,\cdot\,) - f(\cdot+ h)$, $ h \in G$. 8. We show that $L$ coincides with the real space $L_{2}^{0}(G)$. Otherwise there exists a nontrivial real function $r \in L_{2}^{0}(G)$ such that
$$
\begin{equation*}
\int_{G}(f(x+h)-f(x))r(x)\,dm(x) \equiv 0 \quad\!\Longrightarrow\!\quad \int_{G}f(x+h)r(x)\,dm(x) \equiv \mathrm{const}, \quad h \in G.
\end{equation*}
\notag
$$
As $r$ is real valued, its Fourier expansion looks like
$$
\begin{equation*}
r(x)=\sum_{\nu=1}^{\infty}d_{\nu} \chi_{\nu}(x)+\sum_{\nu=1}^{\infty} \overline{d_{\nu}} \overline{\chi}_{\nu}(x), \qquad d_{\nu} \in \mathbb{C}.
\end{equation*}
\notag
$$
Now by the expansion
$$
\begin{equation*}
f(x+h)=\sum_{\nu=1}^{\infty}c_{\nu}\chi_{\nu}(h)\chi_{\nu}(x)+ \sum_{\nu=1}^{\infty}c_{\nu}\overline{\chi}_{\nu}(h)\overline{\chi}_{\nu}(x)
\end{equation*}
\notag
$$
we have
$$
\begin{equation*}
\sum_{\nu=1}^{\infty}c_{\nu}\overline{d_{\nu}}\chi_{\nu}(h)+ \sum_{\nu=1}^{\infty}c_{\nu}d_{\nu}\overline{\chi}_{\nu}(h) \equiv \mathrm{const}, \qquad h \in G.
\end{equation*}
\notag
$$
Since the sequences $\{c_{\nu} \}_{\nu \in \mathbb{N}}$ and $ \{d_{\nu} \}_{\nu \in \mathbb{N}}$ belong to $l_{2}$, the left-hand side of the above identity is an absolutely convergent Fourier series in $h$, and therefore $c_{\nu}d_{\nu}= c_{\nu}\overline{d_{\nu}}=0$, $ \nu \in \mathbb{N}$. Since $c_{\nu} > 0$ for $\nu \in \mathbb{N}$, we obtain $d_{\nu}=0$ for $\nu \in \mathbb{N}$, that is, $r\equiv 0$, which contradicts the assumptions. Thus, the subspace $L$, and therefore the subgroup $S$, coincides with the real space $L_{2}^{0}(G)$. Theorem 1 is proved. § 4. Complex caseRemark 1. Let $G$ be a nontrivial compact Abelian group. Then there does not exist a function $f$ in the complex space $L_{2}^{0}(G)$ whose sums of shifts (1.1) are dense in this space. In fact, given a disconnected compact Abelian group $G$ or a nontrivial compact Abelian group $G$ whose character group $G^{*}$ is not infinite countable, if there exists a function $f$ such that the sums of shifts of $f$ are dense in the complex space $L_{2}^{0}(G)$, then the sums of shifts of $\operatorname{Re} f$ are dense in the real space $L_{2}^{0}(G)$, in contradiction to Lemma 3 or Lemma 4, respectively. Now let $G$ be a connected compact Abelian group with infinite countable character group $G^{*}=\Gamma_{1} \sqcup \Gamma_{2} \sqcup \{\mathbb{I} \}$, where $ \Gamma_{1}=\{\chi_{\nu} \}_{\nu=1}^{\infty}$ and $ \Gamma_{2}=\{\overline{\chi}_{\nu} \}_{\nu=1}^{\infty}$, and assume that there exists a function $f$ in the complex space $L_{2}^{0}(G)$ such that the sums of shifts of $f$ are dense in this space. Let $m_{G}$ and $ m_{G^{*}}$ be the Haar measures on $G$ and $G^{*}$, respectively. Then the Fourier transform
$$
\begin{equation*}
g(\chi):=\widehat{f}(\chi)=\int_{G}f(x)\chi(-x)\,dm_{G}(x)
\end{equation*}
\notag
$$
of $f$ and also the function
$$
\begin{equation*}
g_{1}\colon G^{*} \to \mathbb{C}, \qquad g_{1}(\chi) := \begin{cases} g(\overline{\chi}), &\chi \in \Gamma_{1}, \\ -g(\overline{\chi}), &\chi \in \Gamma_{2}, \\ 0, &\chi \equiv \mathbb{I}, \end{cases}
\end{equation*}
\notag
$$
belong to $L_{2}(G^{*})$ (see [8], Ch. 1, vol. 1.6.1), and the inverse Fourier transform $f_{1}:=\check{g_{1}}$ belongs to $L_{2}^{0}(G)$ because
$$
\begin{equation*}
\int_{G}f_{1}(x)dm_{G}(x)= \widehat{f_{1}}(\mathbb{I})=g_{1}(\mathbb{I})=0.
\end{equation*}
\notag
$$
Note that by assumption the function $f$ is nontrivial, so that $g, g_{1}$ and $ f_{1}$ are also nontrivial. For $y \in G$ we have
$$
\begin{equation*}
\begin{aligned} \, &\int_{G} f(x+y)\overline{f_{1}(x)}\,dm_{G}(x)=\int_{G^{*}} \widehat{f(\cdot+y)}(\chi) \overline{\widehat{f_{1}(\,\cdot\,)}(\chi)}\,dm_{G^{*}}(\chi) \\ &\qquad =\int_{G^{*}} \chi(y) \widehat{f}(\chi) \overline{g_{1}(\chi)}\,dm_{G^{*}}(\chi) =\int_{G^{*}} \chi(y) g(\chi) \overline{g_{1}(\chi)}\,dm_{G^{*}}(\chi) \\ &\qquad =\int_{\Gamma_{1}} \chi(y) g(\chi) \overline{g(\overline{\chi})}\,dm_{G^{*}}(\chi) - \int_{\Gamma_{2}} \chi(y) g(\chi) \overline{g(\overline{\chi})}\,dm_{G^{*}}(\chi) \\ &\qquad =\int_{\Gamma_{1}} \chi(y) g(\chi) \overline{g(\overline{\chi})}\,dm_{G^{*}}(\chi) - \int_{\Gamma_{1}} \overline{\chi}(y) g(\overline{\chi}) \overline{g(\chi)}\,dm_{G^{*}}(\chi) \\ &\qquad =\int_{\Gamma_{1}} \bigl( \chi(y) g(\chi) \overline{g(\overline{\chi})} - \overline{\chi(y) g(\chi) \overline{g(\overline{\chi})}} \bigr)\,dm_{G^{*}}(\chi). \end{aligned}
\end{equation*}
\notag
$$
Thus,
$$
\begin{equation*}
\operatorname{Re} \int_{G} f(x+y)\overline{f_{1}(x)}\,dm_{G}(x)=0
\end{equation*}
\notag
$$
for all $y \in G$, and thus sums of shifts of $f$ lie in a real hyperplane in the complex space $L_{2}^{0}(G)$, so that they cannot be dense in this space. The author is grateful to P. A. Borodin for stating the problem and making useful comments. 
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\Bibitem{Dyu24} |
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