\section{Preliminaries}\label{S1}


Denote by $S(\R^d)$ the Schwartz space of test functions $\p\in C^\infty(\R^d)$ with the finite norms
 $$
  N_n(\p)=\sup_{\R^d}\max_{\|k\|\le n}\,(1+|x|)^n |D^k\p(x)|,\quad n=0,1,2,\dots,
 $$
where
$$
|x|=(x_1^2+\dots+x_d^2)^{1/2},\   k=(k_1,\dots,k_d)\in(\N\cup\{0\})^d,\ \|k\|=k_1+\dots+k_d,\  D^k=\partial^{k_1}_{x_1}\dots\partial^{k_d}_{x_d}.
 $$
 These norms generate the topology on $S(\R^d)$.  Elements of the space $S^*(\R^d)$ of continuous linear functionals on $S(\R^d)$ are called {\it temperate distributions}.
The Fourier transform of a temperate distribution $f$ is defined by the equality
\begin{equation}\label{a0}
\hat f(\p)=f(\hat\p)\quad\mbox{for all}\quad\p\in S(\R^d),
\end{equation}
where
$$
   \hat\p(y)=\int_{\R^d}\p(x)\exp\{-2\pi i\la x,y\ra\}dx
 $$
is the Fourier transform of the function $\p$. Also,
$$
   \check\p(y)=\int_{\R^d}\p(x)\exp\{2\pi i\la x,y\ra\}dx
 $$
 means the inverse Fourier transform. Note that the Fourier transform is an isomorphism of $S(\R^d)$ on $S(\R^d)$ and, respectively, $S^*(\R^d)$ on $S^*(\R^d)$.

  We will say that a set $A\subset\R^d$ is {\it locally finite} if the intersection of $A$ with any ball is finite,  $A$ is {\it relatively dense} if there is $R<\infty$
  such that $A$ intersects with each ball of radius $R$, and $A$ is {\it uniformly discrete}, if $A$ is locally finite and has a strictly positive separating constant
 $$
 \eta(A):=\inf\{|x-x'|:\,x,\,x'\in A,\,x\neq x'\}.
  $$
  A locally finite set $A$ is of {\it bounded density} if
  $$
        \sup_{x\in\R^d}\# A\cap B(x,1)<\infty.
  $$
  As usual, $\# E$ means a number of elements of the finite set $E$, and $B(x,r)$ means the ball with center in $x$ and radius $r$.

  An element $f\in S^*(\R^d)$ is called {\it a crystalline measure} if $f$ and $\hat f$ are  complex-valued measures on $\R^d$ with locally finite supports.
  The support of $\hat f$ for a  distribution $f\in S^*(\R^d)$ is called {\it spectrum} of $f$.

  Denote by $|\mu|(A)$ the variation of a complex-valued measure $\mu$ on $A$. If both measures $|\mu|$ and $|\hat\mu|$ have  locally finite supports and belong to $S^*(\R^d)$,
we say that $\mu$ is a {\it Fourier quasicrystal}. A measure $\mu=\sum_{\l\in\L}a_\l\d_\l$ with $a_\l\in\C$ and countable $\L$ is called {\it purely point}.


  Furthermore, it is well known that every distribution with locally finite support $\L$ has the form
 \begin{equation}\label{r0}
  f=\sum_{\l\in\L}\sum_k p_k(\l)D^k\d_\l,\quad k\in(\N\cup\{0\})^d,
  \end{equation}
 where the internal sum is finite for every $\l\in\L$, and $\d_\l$ is the unit mass at the point $\l$.
\begin{Pro}\label{P1}
i) If a distribution $f\in S^*(\R^d)$ has locally finite support, then
  \begin{equation}\label{r1}
f=\sum_{\l\in\L}\sum_{\|k\|\le K}p_k(\l)D^k\d_\l,\quad k\in(\N\cup\{0\})^d,
\end{equation}
where $K<\infty$ does not depend on $\l$ (\cite{F1}, Proposition 1),

ii) if a distribution $f\in S^*(\R^d)$ has uniformly discrete support, then $K$ does not depend on $\l$  and for some $M<\infty$ and all $k$
\begin{equation}\label{r2}
p_k(\l)=O(|\l|^M)\quad\mbox{as}\quad \l\to\infty,
\end{equation}
(\cite{P}, Proposition 1),

iii) if a distribution $f\in S^*(\R^d)$  of form \eqref{r1}  with locally finite support $\L$ and locally finite spectrum $\G$  has the property
\begin{equation}\label{r3}
\sum_{|\l|\le r}\sum_{\|k\|=0}^K|p_k(\l)|=O(r^{d+M})\phantom{XXXXXX}(r\to\infty)
\end{equation}
with $M\ge0$, then
\begin{equation}\label{m}
\hat f=\sum_{\g\in\G}\sum_{\|m\|\le M} q_m(\g)D^m\d_\g,\quad m\in(\N\cup\{0\})^d,
\end{equation}
If, in addition, $M$ is an integer, then for $\|m\|=M$
\begin{equation}\label{r}
q_m(\g)=O(|\g|^K)\quad\mbox{as}\quad \g\to\infty.
\end{equation}
 with the same $K$ as in \eqref{r1}. If $M$ is a positive real number, and $\G$ is uniformly discrete, then \eqref{r} is valid for all $m$ (\cite{F1}, Proposition 3),

In particular, if $f$ is a measure with uniformly bounded masses, support of bounded density, and locally bounded spectrum, then $K=M=0$ and $\hat f$ is a measure with uniformly bounded masses as well.
\end{Pro}
{\bf Remark 1}. Analysis of the proof of the Proposition shows that if  we replace $O(r^{d+M})$ by $o(r^{d+M})$ in \eqref{r3},
then we may replace the inequality $\|m\|\le M$  by $\|m\|<M$ in \eqref{m}.
\medskip

{\it A lattice} is a discrete (locally finite) subgroup of $\R^d$. If $A$ is a lattice or a coset of some lattice in $\R^d$, then its rank $r(A)$ is the dimension of the smallest
translated subspace of $\R^d$ that contains $A$.  Every lattice $L$ of rank $k$ has the form $T\Z^k$, where $T:\,\Z^k\to\R^d$ is a linear operator of rank $k$. For $k=d$ we get a  full-rank lattice.
In this case the lattice
$$
L^*=\{y\in\R^d: <\l,y>\in\Z\quad \forall \l\in L\}
$$
 is called the conjugate  lattice.  It follows from Poisson's formula
 $$
 \sum_{n\in\Z^d}f(n)=\sum_{n\in\Z^d}\hat f(n),\qquad f\in S(\R^d),
 $$
   that for a full-rank lattice $L=T\Z^d$ we have
 \begin{equation}\label{a1}
      \widehat{\sum_{\l\in L} \d_\l}=|\det T|^{-1}\sum_{\l\in L^*} \d_\l.
 \end{equation}


     \section{Introduction, one-dimensional case}\label{S2}



We begin with the following result of N.Lev and A.Olevskii \cite{LO1}:
\begin{Th}\label{T1}
Let $\mu$ be a crystalline measure on $\R$ with uniformly discrete support and spectrum. Then $\supp\mu$ is a subset of a finite union of translates of a single lattice $L\subset\R$
(i.e.,of arithmetic progressions with the same difference). Moreover, $\mu$ has the form
$$
   \mu=\sum_{j=1}^N \sum_{\l\in L+\tau_j}P_j(\l)\d_\l
$$
for certain reals $\tau_j$ and trigonometric polynomials $P_j(\l)$ ($1\le j\le N$).
\end{Th}
Theorem \ref{T1} remains valid under a weaker assumption that $\supp\mu$ is a relatively dense set of bounded density (not assumed to be uniformly discrete) \cite{LO2}. However there exist
 examples of crystalline measures on $\R$, whose supports  are not contained in any finite union of translates of a lattice (see for example \cite{LO3}, \cite{M2}, \cite{KS}).

In article \cite{LR} Theorem \ref{T1} was extended to temperate distributions on the line:
\begin{Th}\label{T2}
Let $f$ be a temperate distribution on $\R$ such that $\L=\supp f$ and
$\G=\supp\hat f$ are uniformly discrete sets. Then $f$ can be represented in the form
$$
f=\sum_{\tau,\omega, l, p}c(\tau,\omega, l, p)\sum_{\l\in L}\l^le^{2\pi i\l\omega}\d_{\l+\tau}^{(p)},
$$
where $L$ is a lattice on $\R$, $(\tau,\omega, l, p)$ goes through a finite set of quadruples such that $\tau,\omega$ are
real numbers, $l, p$ are nonnegative integers, and $c(\tau,\omega, l, p)$ are complex numbers.

The result is still correct if the support $\L$ has bounded density and the coefficients $p_k(\l)$ in \eqref{r0} satisfy \eqref{r2}, while the spectrum $\G$ is
uniformly discrete.
\end{Th}


\section{Introduction, multivariate case}\label{S3}


Various analogs of Theorem \ref{T2} for temperate distributions on $\R^d$ were obtained in \cite{P} under conditions of locally finite set of differences $\L-\L$ and $\G-\G$ and in \cite{F1}
under conditions of locally finite  $\L-\L$, not too fast approaching points from $\G$, and uniformly separated from zero and infinity coefficients $p_k(\l)$.
\smallskip

In \cite{LO1}, \cite{LO2}  N.Lev and A.Olevskii  proved the following result:
\begin{Th}\label{T3}
Let $\mu$ be a {\bf positive}  measure on $\R^d$ such that $\L=\supp\mu$ and
$\G=\supp\hat\mu$ are uniformly discrete sets. Then $\L$ is contained in a finite union of translates of a lattice of rank $d$.  The same is valid if $\L$ is locally finite and of bounded density not assumed to be uniformly discrete.
\end{Th}
It was proved in \cite{F2} that there is a signed measure on $\R^2$ such that its support and  spectrum
are both uniformly discrete and simultaneously are unions of two incommensurable full-rank lattices. Therefore neither support, nor spectrum can be finite unions of translations of a {\bf single} lattice.

But there are results of a different type
 for  measures in $\R^d$ with discrete support, in which a {\bf finite number of  lattices} are already involved (\cite{M3},  \cite{KL}, \cite{K1}, \cite{Co}).
 In fact, the following result was proved:
\begin{Th}\label{T4}
 Let  $\mu$ be a measure on $\R^d$ with uniformly discrete support $\L$. If complex masses $\mu(\{\l\})$ at  points $\l\in\L$  take  values only from a finite set $F\subset\C$,
 and the measure $\hat\mu$ be purely point and satisfies the condition
 \begin{equation}\label{mu}
|\hat\mu|(B(0,r))=O(r^d) \quad (r\to\infty),
\end{equation}
 then $\L$  is a finite union of translations of several, possibly incommensurable, full-rank lattices.
\end{Th}
In the paper \cite{LR}, the following problem was posed:
 \smallskip

{\bf Problem 2.} Let $\mu$ be a measure on $\R^d$ with discrete support $\L$ and its Fourier transform in the sense of distributions $\hat\mu$ is a measure with discrete support as well.
Does it follow that $\L$ can be covered by a finite union of translates of several lattices?
\smallskip

Remark that the periodic structure of a crystalline measure in the multidimensional case  follows under extra assumptions on  masses (\cite{F4},\cite{F5}):
\begin{Th}\label{T5}
 Let $\mu$ be a measure on $\R^d$ with uniformly discrete support $\L$ such that $\inf_{\l\in\L}|\mu(\{\l\})|>0$ and the measure $\hat\mu$  is purely point and satisfy \eqref{mu}.
 Then  $\L$  is a finite union of translates of several disjoint full-rank lattices.

  Moreover,  there exist an integer $J$, lattices $L_1,\dots,L_J$ in $\R^d$ of rank $d$ (some of them may coincide), points $\l_1,\dots,\l_J\in\L$,
  a bounded set $\{\a_s^j:\,s\in\N,\,1\le j\le J\}\subset\R^d$, and functions
  $$
  F_j(y)=\sum_s b_s^j e^{2\pi i\la y,\a_s^j\ra}\quad\mbox{with}\quad \sum_s|b_s^j|<\infty,\quad j=1,\dots,J,
  $$
   such that
 $$
 \mu=\sum_{j=1}^J F_j(\l)\sum_{\l\in L_j+\l_j}\d_\l.
 $$
 \end{Th}
 The proofs of theorems \ref{T4} and \ref{T5} are based on Cohen's Idempotent Theorem (see, e.g., \cite{Ru};
 the first using of the Idempotent Theorem to prove structure of tilings was in \cite {LM}):
\begin{Th}\label{T6}
Let $G$ be a locally compact Abelian group and $\hat G$ its dual group. If $\nu$ is a finite Borel measure on $G$ and is such that
its Fourier transform $\hat\nu(\l)$ takes only  values $0$ and $1$, then the set $\{\l:\,\hat\nu(\l)=1\}$ belongs to the coset ring of $\hat G$.
\end{Th}
Recall that the coset ring of an Abelian topological group $G$ is the smallest collection  of subsets of $G$ that is closed under complement, finite unions, and finite intersections,
  which contains all cosets of all open subgroups of $G$. In particular, if $G=\R^d_{discr}$, i.e., the space $\R^d$ with respect to the discrete topology, then the coset ring contains all cosets of all subgroups of $G$.
 \begin{Th}\label{T7}[M.Kolountzakis \cite{K1}, Theorem 3]
  All  elements of the coset ring for $\R^d_{discr}$, which are discrete in the Euclidean topology of $\R^d$, are finite unions of sets of the type $A\setminus(\cup_{j=1}^J B_j)$,
where  $A,\ B_j$ are discrete cosets in the Euclidean topology.
\end{Th}


\section{Results}\label{S4}

Here we get the following extension of Theorems \ref{T3} and \ref{T5} to temperate distributions:
\begin{Th}\label{T8}
Suppose that a temperate distribution $f$ has   locally finite support $\L$ of  bounded density, a uniformly discrete spectrum $\G$, and all coefficients $p_k(\l)$ in \eqref{r0} are
uniformly bounded and non-negative. Then $\L$ is contained in a finite union of translates of several full-rank lattices.

The non-negativity of  coefficients $p_k(\l)$  can be replaced by the following: for each $k\in(\N\cup\{0\})^d$ there are $\xi_k\in\R$ such that $\Arg\,p_k(\l)=\xi_k$
for all $p_k(\l)\neq0,\,\l\in\L$.
\end{Th}
\begin{Th}\label{T9}
Suppose that a temperate distribution $f$ of form \eqref{r0} has both uniformly discrete support $\L$ and spectrum $\G$.
If there are constants $c,\,C$ such that for  coefficients in \eqref{r0}  the inequalities  hold
\begin{equation}\label{c3}
 0<c\le\sum_k |p_k(\l)|\le C<\infty,\qquad \forall\ \l\in\L,
 \end{equation}
then $\L$ is  a  finite union of translations of several full-rank lattices.
More accurately, there exist integers $J,\,K$, lattices $L_1,\dots,L_J$ in $\R^d$ of rank $d$ (some of them may coincide), points $\l_1,\dots,\l_J\in\L$, and
a bounded set of exponents $\b _s=\b_s(j,k)\in\R^d$  such that
$$
f=\sum_{j=1}^J\sum_{\|k\|\le K}\sum_{\l\in L_j+\l_j} F_{k,j}(\l)D^k\d_\l,
$$
where
$$
F_{k,j}(x)=\sum\limits_s b_s(k,j) e^{2\pi i\la x,\b_s\ra}\quad\mbox{with}\quad\sum\limits_s|b_s(k,j)|<\infty.
$$
\end{Th}
{\bf Remark 2}.  Proposition \ref{P1} i) implies that $f$ has form \eqref{r1} with $K<\infty$, therefore we may replace the sum $\sum_k|p_k(\l)|$ by $\sup_k|p_k(\l)|$ in \eqref{c3}.
Also, we can replace the right-hand bound in \eqref{c3} by
$$
p_k(\l)=o(|\l|)\quad (\l\to\infty)\qquad\forall k,
$$
because in this case we get again $p_k(\l)=O(1)$ (see Remark 3 in Section \ref{S6}).

Nevertheless,  some upper and lower bounds are required.

Let $A=\{n\in\Z^2:\,n_1\neq0\}$. It follows from \eqref{a0} and \eqref{a1} that the Fourier transform of the measure with unbounded coefficients
 $$
   \sum_{n\in A} n_1\d_n=\sum_{x\in\Z^2}x_1\d_x
  $$
  is the distribution
$$
   \frac{i}{2\pi}\sum_{\l\in\Z^2}\frac{\partial}{\partial\l_1}\d_\l.
$$
Also, the Fourier transform of the measure whose masses do not separate from zero
$$
   \sum_{n\in A} \sin\left(\frac{2\pi n_1}{\sqrt{5}}\right)\d_n=\frac{1}{2i}\sum_{x\in\Z^2}\left(e^{2\pi ix_1/\sqrt{5}}-e^{-2\pi ix_1/\sqrt{5}}\right)\d_x
 $$
 is equal to
 $$
   \frac{1}{2i}\sum_{y\in\Z^2}\d_{y+\a}-\frac{1}{2i}\sum_{y\in\Z^2}\d_{y-\a},\qquad \a=(1/\sqrt{5},0),
$$
hence in both cases the Fourier transforms have uniformly discrete supports. But $A$  is not  a finite union of cosets of several full-rank lattices, because in the opposite case the set $\Z\setminus\{0\}$,
which is the projection of $A$ on $\R_{x_1}$, would be  a finite union of arithmetical progressions, that is impossible.
\smallskip

But for some distributions  with unbounded coefficients their supports are {\bf subsets} of a finite union of translations of full-rank lattices:
\begin{Th}\label{T10}
Let $f\in S^*(\R^d)$ of form \eqref{r0} have both uniformly discrete support $\L$ and spectrum $\G$.
If there are integers $h(k)\in\N\cup\{0\}$  and constants $c,\,C$ such that for coefficients in \eqref{r0}  the inequalities  hold
\begin{equation}\label{c0}
 0<c\le\max_k|p_k(\l)|(1+|\l|)^{-h(k)}\le C<\infty,\qquad \forall\ \l\in\L,
 \end{equation}
then $\L$ is a subset of a  finite union of translations of several full-rank lattices that can be incommensurable.
\end{Th}


The above theorems  are based on the following  result for  temperate distributions with uniformly discrete spectrum, which is also of independent interest:
\begin{Th}\label{T11}
 Suppose $f\in S^*(\R^d)$  of  form \eqref{r0} has  locally finite support $\L$ of bounded density and uniformly discrete spectrum $\G$. Set for  $k\in(\N\cup\{0\})^d$
 \begin{equation}\label{c2}
\mu_k=\sum_{\l\in\L}p_k(\l)\d_\l.
\end{equation}

a) If $p_k(\l)$ are uniformly bounded in $k\in(\N\cup\{0\})^d$ and $\l\in\L$, then  $\hat\mu_k\in S^*(\R^d)$ are measures of the form
$$
 \hat\mu_k=\sum_{\g\in\G_k}q_k(\g)\d_\g
$$
 with uniformly bounded masses $q_k(\g)$  and uniformly discrete supports $\G_k$ such that
\begin{equation}\label{et}
\eta(\G_k)\ge\eta(\G)\qquad \forall k.
\end{equation}

b) if both $\L$ and $\G$ are uniformly discrete, then $\hat\mu_k\in S^*(\R^d)$ are distributions of the form
\begin{equation}\label{m2}
\hat\mu_k=\sum_{\|m\|\le M}\sum_{\g\in\G_k}q_{k,m}(\g)D^m\d_\g
\end{equation}
with uniformly bounded coefficients $q_{k,m}(\g)$ and uniformly discrete supports $\G_k$, which satisfy \eqref{et}
\end{Th}
Since $\mu_k$ is the inverse Fourier transform of $\hat\mu_k$, we see that $\mu_k\in S^*(\R^d)$. Also, by Proposition \ref{P1} i), $\mu_k\equiv0$ for $\|k\|$ large enough. Hence,
\begin{equation}\label{f2}
   f=\sum_k D^k\mu_k,\phantom{XXXXX} \hat f(y)=\sum_k (2\pi i)^{\|k\|}y^k\hat\mu_k,
 \end{equation}
where, as usually, $y^k=y_1^{k_1}\cdot\dots\cdot y_d^{k_d}$.



\section{Almost periodic functions and distributions}\label{S5}


We recall here  definitions and properties of almost periodic functions and distributions that will be used in what follows. A more complete exposition of these issues is available in \cite{C},
\cite{A}, \cite{M1}, \cite{M2}, \cite{R}.

\begin{De}\label{D1} A continuous function $g$ on $\R^d$ is  almost periodic if for any  $\e>0$ the set of its $\e$-almost periods
 $$
  \{\tau\in\R^d:\,\sup_{t\in\R^d}|g(t+\tau)-g(t)|<\e\}
 $$
is a relatively dense set in $\R^d$.
\end{De}
An equivalent definition follows:
\begin{De}\label{D2} A continuous function $g$ on $\R^d$ is  almost periodic if for any sequence $\{t_n\}\subset\R^d$ there is a subsequence  $\{t'_n\}$ such that
 the sequence of functions $g(t+t'_n)$ converge uniformly in $t\in\R^d$.
\end{De}
Using an appropriate definition, one can prove various properties of almost periodic functions.
\begin{itemize}
\item almost periodic functions are  bounded and uniformly continuous on $\R^d$,
\item the class of almost periodic functions is closed with respect to taking absolute values and linear combinations of a finite family of functions,
 \item a limit of a uniformly convergent sequence  of almost periodic functions is also almost periodic,
 \item any finite family of almost periodic functions has a relatively dense set of common $\e$-almost periods,
\item  for any almost periodic function $g(x)$ on $\R^d$ the function $h(t)=g(tx)$ is almost periodic in $t\in\R$ for any fixed $x\in\R^d$; in particular,
$g(t_1,\dots,t_d)$ is almost periodic in each variable $t_j\in\R,\ j=1,\dots,d$, if the other variables are held fixed.
 \end{itemize}
Also, we will use the following definition
\begin{De} A distribution $g$ is  almost periodic if the function $(g(y),\p(t-y))$
is almost periodic in $t\in\R^d$ for each $C^\infty$-function $\p$ on $\R^d$ with compact support.
A measure $\mu$ is almost periodic if it is almost periodic as a distribution from $S^*(\R^d)$.
 \end{De}
Clearly, every linear combination of almost periodic distributions is almost periodic, and each almost periodic distribution  has a relatively dense support.

 Note that the usual definition of almost periodicity for measures (instead of $\p\in C^\infty$, we consider  continuous $\p$ with compact support)
 differs from the one given above. However these definitions coincide for nonnegative measures or measures with uniformly discrete support (see  \cite{A}, \cite{M1}, \cite{F2}).
\begin{Pro}\label{P2}
Let $\mu=\sum_{\l\in\L} p(\l)\d_\l,\  p(\l)\in\C$, be an almost periodic measure with uniformly discrete support $\L\subset\R^d$, then

i) the masses $p(\l)$ are uniformly bounded,

ii)  the measure $|\mu|=\sum_{\l\in\L}|p(\l)|\d_\l$ is almost periodic,

iii) if $\inf_{\l\in\L}|p(\l)|\ge\a>0$, then the measure $\d_\L:=\sum_{\l\in\L}\d_\l$ is almost periodic as well,

iv) for any $\e>0$ and $\l\in\R^d$ there is a relatively dense set $T$ such that $|\mu(\l')-\mu(\l)|<\e$ for all $\l'\in T$,

v) if $\mu_n,\,n=1,\dots,N$ be almost periodic measures  with uniformly discrete supports, then for any vector $\l\in\R^d$ and numbers $\e>0,\ \rho>0$ there is a relatively dense set $T$ such that
for every $\l'\in T$ and some $\l^{(n)}\in B(\l',\rho)$
\begin{equation}\label{m1}
|\mu_n(\l^{(n)})-\mu_n(\l)|<\e,\quad n=1,\dots,N.
\end{equation}
 \end{Pro}

  The condition on $\L$ to be uniformly discrete is essential (see\cite{FK}).
\smallskip

 {\bf Proof of Proposition \ref{P2}}. i) Let $\p$ be $C^\infty$ function with support in $B(0,\eta(\L)/2)$ such that $\p(0)=1$. The function $(\mu(y),\p(t-y))$ is almost periodic,
 hence it is uniformly bounded, and the numbers $p(\l)=(\mu(y),\p(\l-y))$ are uniformly bounded as well.

 ii) Every $C^\infty$-function  with compact support is a finite linear combination of shifts of nonnegative $C^\infty$-functions with supports in balls of radius $\eta/2$
 \footnote{Using Theorem on Partition of Unity, it suffices to prove this for the case of a real-valued  $\p\in C^\infty$ with support in the ball $B(x,\eta(\L)/4)$.
 Then we have $\p=\psi-(\psi-\p)$, where $\psi$ is an arbitrary $C^\infty$ nonnegative function with support in $B(x,\eta(\L)/2)$ such that $\p\le \psi$.}.
  Consequently we can check the almost periodicity  using only  such functions.

 Let $\p$ be one of them and $\supp\p\subset B(x,\eta/2)$. For each $t,\tau\in\R^d$ we have
 $$
       |(|\mu|(y),\p(t-y))-(|\mu|(y),\p(t+\tau-y))|=\left|\sum_{\l\in\L}\p(t-\l)|p(\l)|-\sum_{\l'\in\L}\p(t+\tau-\l')|p(\l')|\right|.
 $$
 Evidently, here each sum consists of at most one nonzero term. Therefore, for some $\l, \l'\in\L$ that depends on $t,\tau$ we get
 $$
       |(|\mu|(y),\p(t-y))-(|\mu|(y),\p(t+\tau-y))|=|\p(t-\l)|p(\l)|-\p(t+\tau-\l')|p(\l')||
 $$
  $$
   \le  |\p(t-\l)p(\l)-\p(t+\tau-\l')p(\l')|=|(\mu(y),\p(t-y))-(\mu(y),\p(t+\tau-y))|.
  $$
 Hence every $\e$-period $\tau$ of the function $(\mu(y),\p(t-y))$ is an $\e$-period of the function $(|\mu|(y),\p(t-y))$.

 iii) Pick $\e>0$ and $\b<\min\{\a,\eta(\L)/2\}$ such that $|\p(x)-\p(x')|<\e$ for $|x-x'|<\b$. Let $\psi$ be $C^\infty$-function such that
$\supp\psi(x)\subset B(0,\b)$ and $\psi(0)=1$.  Let  $\tau$ be some $\b$-almost period of the function $(\mu(y),\psi(t-y))$, and let $t\in B(\l,\b)$ for $\l\in\L$,
Since $\b<\eta(\L)/2$, we get for at most one $\l'\in\L$ that depends on $t+\tau$
   $$
      \b>\left|\sum_{\l\in\L}\psi(t-\l)p(\l)-\sum_{\l'\in\L}\psi(t+\tau-\l')p(\l')\right|= |p(\l)\psi(t-\l)-p(\l')\psi(t+\tau-\l')|.
   $$
For $t=\l$ we get
    $$
    |p(\l)-p(\l')\psi(\l+\tau-\l')|<\b.
    $$
 The inequality $\b<\a\le|p(\l)|$ implies that $\psi(\l+\tau-\l')\neq0$. Hence, $|\l+\tau-\l'|<\b$. Therefore, $|\p(t-\l)-\p(t+\tau-\l')|<\e$ and
  $$
       \left|\sum_{\l\in\L}\p(t-\l)-\sum_{\l'\in\L}\p(t+\tau-\l')\right|=|\p(t-\l)-\p(t+\tau-\l')|<\e.
  $$
This inequality is valid for all $\b$-almost periods $\tau$ from a relatively dense set. Thus the function $(\d_\L(y),\p(t-y))$ is almost periodic.

 iv) It is enough to consider the case $\e\le|\mu(\l)|$. Let $\rho<\eta(\supp\mu)/2$, and let $\p$ be $C^\infty$ function on $\R^d$ such that
 $$
   0\le\p(t)\le1,\quad \p(0)=1,\quad \p(t)=\p(-t),\quad \supp\p\subset B(0,\rho).
 $$
We have $\mu(\l)=(\p(\l-x),\mu(x))$.  If $\tau$ be an $\e/3$-almost period of the almost periodic function $(\p(t-x),\mu(x))$, we get
$$
|(\p(\l+\tau-x),\mu(x))-\mu(\l)|<\e/3.
$$
Therefore, there is a single point $\l'\in B(\l+\tau,\rho)$ such that
$$
|\mu(\l')\p(\l+\tau-\l')-\mu(\l)|<\e/3.
$$
We get $\l'\in\supp\mu$ and $|\mu(\l')|>2\e/3$. Since $-\tau$ is an $\e/3$-almost period as well, we see that for some $\l''\in B(\l'-\tau,\rho)$
$$
|\mu(\l'')\p(\l'-\tau-\l'')-\mu(\l')|<\e/3.
$$
We get $\l''\in\supp\mu$ and  $|\l''-\l|<2\rho$, hence, $\l''=\l$. Then we have for $t=\l+\tau-\l'$
$$
 |\mu(\l)-\mu(\l')|\le|\mu(\l)-\p(t)\mu(\l')|+|\mu(\l')-\p(t)\mu(\l')|\le|\mu(\l)-\p(t)\mu(\l')|+(1-\p^2(t))|\mu(\l')|
$$
$$
\le|\mu(\l)-\p(t)\mu(\l')|+|\mu(\l')-\p(-t)\mu(\l)|+|\p(t)\mu(\l)-\p^2(t)\mu(\l')|<\e.
$$
Recall that almost periods form a relatively dense set.

 v) The proof is very close to the previous one. We take $\rho<\min_n \eta(\mu_n)/2$,  the same function $\p$, and $\l'=\l+\tau$,
 where $\tau$ is a common $\e/3$-almost period of the functions $(\p(\l-x),\mu_n(x))$. Then, as $\l^{(n)}$ we take points satisfying  the conditions
$$
|\mu_n(\l^{(n)})\p(\l'-\l^{(n)})-\mu_n(\l)|<\e/3.
$$
Further, arguing as above, we obtain \eqref{m1}. \bs
\medskip

Typical examples of almost periodic functions on $\R^d$ are  sums of the form
\begin{equation}\label{W}
   f(t)=\sum_n a_n e^{2\pi i\la t,s_n\ra},\quad a_n\in\C,\quad s_n\in\R^d,\quad \sum_n|a_n|<\infty.
\end{equation}
It is not hard to prove that $\hat f=\sum_n a_n\d_{s_n}$.

Denote by $W$ the class of functions admitting  representation \eqref{W}. It is easy to check that a product of two functions from $W$ belongs to $W$ as well.
Also, in Section \ref{S7} we will use the following local version of the classical Wiener-Levi Theorem:
\begin{Pro}\label{P3}[\cite{F4}]
 Let  $K\subset\C$ be an arbitrary compact set,  $h(z)$ be a holomorphic function on a neighborhood of $K$, and $f\in W$. Then there is a function $g\in W$ such that if $f(x)\in K$ then $h(f(x))=g(x)$.
 \end{Pro}
Next, the Bohr compactification $\hR$ of $\R^d$ is a compact group with the dual $\R^d_{discr}$, and  $\R^d$ is a dense subset of $\hR$ with respect to the topology on $\hR$. Moreover,
restrictions to $\R^d$ of continuous functions on $\hR$ are almost periodic functions on $\R^d$ (see, e.g.,\cite{Ru}).



\section{Representation of distributions with discrete support and spectrum}\label{S6}

First we prove the following lemma:
\begin{Le}\label{L1}
Let $g_0(y), g_1(y),\dots, g_N(y)$ be temperate distributions on $\R^d$ such that for every $C^\infty$-function $\p(y)$ with compact support the functions
$$
(g_n(y),\p(t-y)),\quad t=(t_1,t_2,\dots,t_d),\quad n=0,\dots,N,
$$
are almost periodic in $t_1\in\R$ for every fixed $(t_2,\dots,t_d)\in\R^{d-1}$. If the distribution
$$
F=\sum_{n=0}^N y_1^n g_n(y),\quad y=(y_1,\dots,y_d),
$$
 has  a uniformly discrete support $T$, then the set $\cup_n\supp g_n$ is uniformly discrete and $\eta(\cup_n\supp g_n)\ge\eta(T)$.
\end{Le}

{\bf Proof}. Note that for every $C^\infty$-function $\p(y)$ with compact support we have
\begin{equation}\label{s1}
   (F(y),\p(t-y))=\sum_{n=0}^\N(g_n(y),y_1^n\p(t-y))=\sum_{m=0}^\N t_1^m \sum_{n=m}^N \P_{n,m}(t),
\end{equation}
where
$$
\P_{n,m}(t)=\binom{n}{m}(g_n(y),(y_1-t_1)^{n-m}\p(t-y)),\quad n\ge m.
$$
We will prove that for every $n,m\le N$ and every $y\in\supp g_n,\,\tilde y\in\supp g_m$ we get $|y-\tilde y|\ge\eta(T)$.

Consider the case $n=m=N$. Assume the converse. Then there are points $y',y''\in\supp g_N$ such that $0<|y'-y''|<\eta(T)$.
Pick
$$
\a<(1/2)\min\{|y'-y''|,\,(\eta(T)-|y'-y''|)\}
$$
 and $C^\infty$-function $\p$ with $\supp\p\subset B(0,\a)$ such that
 $$
\P_{N,N}(y')=(g_N(y),\p(y'-y))\neq0,\qquad \P_{N,N}(y'')=(g_N(y),\p(y''-y))\neq0.
 $$
 Take $\e>0$ such that $|\P_{N,N}(y')|>2\e$, $|\P_{N,N}(y'')|>2\e$. If $\tau$ is a common $\e$-almost period of the functions
 $$
    \P_{N,N}(y'+se_1),\quad \P_{N,N}(y''+se_1),\quad\mbox{where}\quad e_1=(1,0,0\dots 0),\quad s\in\R,
 $$
 then $|\P_{N,N}(y'+\tau e_1)|>\e$, $|\P_{N,N}(y''+\tau e_1)|>\e$. We have
$$
(y'_1+\tau)^{-N}(F(y),\p(y'+\tau e_1-y))=\P_{N,N}(y'+\tau e_1)+\sum_{m=0}^{N-1} (y'_1+\tau)^{m-N} \sum_{n=m}^{N-1}\P_{n,m}(y'+\tau e_1).
$$
Note that all  functions $\P_{n,m}(y'+se_1),\,0\le n,m\le N,$ are almost periodic in $s$, hence they are uniformly bounded on $\R$. Therefore if $\tau$ is large enough then
$$
(y'_1+\tau)^{-N}(F(y),\p(y'+\tau e_1-y))\neq0.
$$
Hence there exists a point $t'\in B(y'+\tau e_1,\a)\cap T$. For the same reason, there exists a point $t''\in B(y''+\tau e_1,\a)\cap T$ with the same $\tau$. We have
\begin{equation}\label{s2}
 |t'-t''|\le|t'-y'-\tau e_1|+|t''-y''-\tau e_1|+|y'-y''|<2\a+|y'-y''|<\eta(T).
\end{equation}
But this inequality is impossible.

Now suppose that our assumption is proved for all $n,\,m$ such that $n+m>M$ with $0\le M<2N$; then we prove it for $p, q\ge0$ such that $p+q=M$.

Assume the converse. There are points $y'\in\supp g_p,\ y''\in\supp g_q$ such that $0<|y'-y''|<\eta(T)$.  Pick
$$
\a<(1/2)\min\{|y'-y''|,\,(\eta(T)-|y'-y''|)\}
$$
and $C^\infty$-function $\p$ with $\supp\p\subset B(0,\a)$ such that
 $$
\P_{p,p}(y')=(g_p(y),\p(y'-y))\neq0,\qquad \P_{q,q}(y'')=(g_q(y),\p(y''-y))\neq0.
 $$
Take $\e>0$ such that $|\P_{p,p}(y')|>2\e$ and $|\P_{q,q}(y'')|>2\e$. Obviously, for a common $\e$-almost period $\tau$ of the functions
$\P_{p,p}(y'+se_1)$ and $\P_{q,q}(y''+se_1)$
  we get
  $$
  |\P_{p,p}(y'+\tau e_1)|>\e,\qquad |\P_{q,q}(y''+\tau e_1)|>\e.
  $$
   If there exists $n>p$ such that at least one of the numbers $\P_{n,j}(y'+\tau e_1),\,j=0,\dots,n$ does not vanish,
 then there is a point $t'\in B(y'+\tau e_1,\a)\cap\supp g_n$. Since $\P_{q,q}(y''+\tau e_1)\neq0$, we see that there is a point $t''\in B(y''+\tau e_1,\a)\cap\supp g_q$. By \eqref{s2}, $|t'-t''|<\eta(T)$, hence
 this inequality contradicts to our assumption $\eta(\supp g_n\cup\supp g_q)\ge\eta(T)$ for $n+q>M$. In the same way, we obtain a contradiction in the case
 when  at least one of the numbers $\P_{n,j}(y''+\tau e_1)$ for $n>q,\,j=0,\dots,n$,  does not vanish.

Therefore, for $t=y'+\tau e_1$ equality \eqref{s1} implies
$$
(y'_1+\tau)^{-p}(F(y),\p(y'+\tau e_1-y))=\P_{p,p}(y'+\tau e_1)+\sum_{m=0}^{p-1} (y'_1+\tau)^{m-p} \sum_{n=m}^{p-1}\P_{n,m}(y'+\tau e_1),
$$
and for $t=y''+\tau e_1$  implies
$$
(y''_1+\tau)^{-q}(F(y),\p(y''+\tau e_1-y))=\P_{q,q}(y''+\tau e_1)+\sum_{m=0}^{q-1} (y''_1+\tau )^{m-q} \sum_{n=m}^{q-1}\P_{n,m}(y''+\tau e_1).
$$
 If $\tau$ is large enough, then we get
$$
(y'_1+\tau)^{-p}(F(y),\p(y'+\tau e_1-y))\neq0,\qquad (y''_1+\tau)^{-q}(F(y),\p(y''+\tau e_1-y))\neq0,
$$
hence there exists $t'\in B(y'+\tau e_1,\a)\cap T$ and $t''\in B(y''+\tau e_1,\a)\cap T$ with the same $\tau$. By \eqref{s2}, we obtain a contradiction. Thus our assumption is valid for all $n,\,m\le N$. \bs

\medskip

{\bf Proof of Theorem \ref{T11}}. It follows from Proposition \ref{P1} i) that there is $K\in\N\cup\{0\}$ such that $\mu_k\equiv0$ for $\|k\|>K$.

In the case a), for any $\p\in S(\R^d)$ and $k$ we have
\begin{equation}\label{c}
(\hat\mu_k(y),\p(t-y))=(\mu_k(x),\check\p(x)e^{-2\pi i\la x,t\ra})=\sum_{\l\in\L}p_k(\l)\check\p(\l)e^{-2\pi i\la\l,t\ra}.
\end{equation}
Since $\check\p\in S(\R^d)$, we get $\check\p(x)=o(|x|^{-d-1})$. Hence,
$$
\sum_{\l\in\L}|p_k(\l)||\check\p(\l)|\le C_1+C_2\sum_{\l\in\L,|\l|>1}|\l|^{-d-1}=C_1+C_2\int_1^\infty s^{-d-1}dn(s),
$$
where $n(s)=\#(\L\cap B(0,s))$. The set $\L$ is of bounded density, therefore, $n(s)=O(s^d)$ and the integral
$$
 \int_1^\infty s^{-d-1}n(ds)=-n(1)+(d+1)\int_1^\infty\frac{n(s)ds}{s^{d+2}}
$$
is finite. Consequently the sum in \eqref{c} is absolutely convergent, and the function $(\hat\mu_k(y),\p(t-y))$ is almost periodic in $t\in\R^d$.

In the case b), by Proposition \ref{P1} ii), we get
\begin{equation}\label{p}
p_k(\l)=O(|\l|^M) \quad\mbox{as}\quad \l\to\infty.
\end{equation}
 Since $\check\p(x)=o(|x|^{-M-d-1})$ for each $H$, we can repeat the previous arguments and obtain
that $\hat\mu_k$ are almost periodic distributions.

Further, by \eqref{f2}, we have
$$
  \hat f=\sum_{k_1=0}^K (2\pi i)^{k_1}y_1^{k_1}g_{k_1}, \quad\mbox{where}\quad g_{k_1}(y)=\sum_{k_2+\dots+k_d\le K-k_1}(2\pi i)^{k_2+\dots+k_d}y_2^{k_2}\dots y_d^{k_d}\hat\mu_{k_1,\dots,k_d},
$$
and for any $\p\in\C^\infty$ with compact support
$$
(g_{k_1}(y),\p(t-y))=\sum_{k_2+\dots+k_d\le K-k_1}(2\pi i)^{k_2+\dots+k_d}(\hat\mu_{k_1,\dots,k_d},y_2^{k_2}\dots y_d^{k_d}\p(t-y)).
$$
Note that each term of the last sum can be rewritten as
$$
  \sum_{m_2\le k_2,\dots,m_d\le k_d}c_{m,k}t_2^{k_2-m_2}\dots t_d^{k_d-m_d}\left[(\hat\mu_k,\,(t_2-y_2)^{m_2}\dots (t_d-y_d)^{m_d}\p(t-y))\right]
$$
with some constants $c_{m,k}$. Since $\mu_k$ are almost periodic distributions, we get that the expressions in square brackets are almost periodic functions in $t\in\R^d$, and hence in $t_1\in\R$.
Therefore the functions $(g_{k_1}(y),\p(t-y))$ are almost periodic in $t_1\in\R$ for any fixed $(t_2,\dots,t_d)\in\R^{d-1}$. Applying  Lemma \ref{L1} to distributions $g_{k_1},\,k_1=0,\dots,K$,
we get  they have uniformly discrete supports and $\eta(\supp g_{k_1})\ge\eta(\G)$.

 For a fixed $k_1$ we have
 $$
 g_{k_1}=\sum_{k_2=0}^{K-k_1}(2\pi i)^{k_2}y_2^{k_2} g_{k_1,k_2},
  $$
where
$$
 g_{k_1,k_2}(y)=\sum_{k_3+\dots+k_d\le K-k_1-k_2}(2\pi i)^{k_3+\dots+k_d}y_3^{k_3}\dots y_d^{k_d}\hat\mu_{k_1,\dots,k_d}.
$$
The functions $(g_{k_1,k_2}(y),\p(t-y))$ are almost periodic in $t_2\in\R$ for any fixed $(t_1,t_3\dots,t_d)\in\R^{d-1}$. Applying  Lemma \ref{L1} to distributions $g_{k_1,k_2},\,k_2=0,\dots,N-k_1$
with respect to the variable $y_2$, we get that these distributions have uniformly discrete supports and $\eta(\supp g_{k_1,k_2})\ge\eta(\G)$.

After a finite number of steps we get that  the support $\G_k$ of the distributions $\hat\mu_k$ for all $k\in(\N\cup\{0\})^d$ are
uniformly discrete and satisfy \eqref{et}. If the masses $p_k(\l)$ are uniformly bounded, then Proposition \ref{P1} iii), yields that $\hat\mu_k$ are measures with uniformly bounded masses.
If masses $p_k(\l)$ satisfy \eqref{p}, we have
$$
  \sum_{|\l|\le r}|p_k(\l)|\le C_1+C_2r^M n(r)=O(r^{M+d}) \quad\mbox{as}\quad r\to\infty.
$$
Applying Proposition \ref{P1} iii) with $K=0$,  we obtain that the coefficients $q_{k,j}(\g)$ in \eqref{m2} are uniformly bounded. \bs
\medskip

{\bf Remark 3}. Let $\L$ be uniformly discrete. If we replace the condition  $p_k(\l)=O(1)$ by $p_k(\l)=o(|\l|)$, then we get for any $\e>0$ and appropriate $r(\e)$
\begin{equation}\label{b1}
  \sum_{|\l|\le r}|p_k(\l)|\le \sum_{|\l|\le r(\e)}|p_k(\l)|+\sum_{r(\e)<|\l|\le r}|p_k(\l)| \le C(\e)+\e r\,n(r).
\end{equation}
Hence, \eqref{b1} is $o(r^{d+1})$ as $r\to\infty$. Applying Proposition \ref{P1} iii) with $K=0,\,M=1$ and taking into account Remark 1,
we obtain that $\hat\mu_k$ (and, of course, $\check\mu_k$) are measures with uniformly bounded masses. Applying
 Proposition \ref{P1} iii) to the measures $\check\mu_k$, we obtain that the coefficients $p_k(\l)$ are uniformly bounded.


\section{Proofs of the main theorems}\label{S7}


{\bf Proof of Theorem \ref{T8}}. By Theorem \ref{T11}, the measures $\mu_k$ from \eqref{c2} have uniformly discrete spectra. Applying Theorem \ref{T3} to these measures,
we get that $\supp\mu_k$ is contained in a finite union of translates of a full-rank lattice.
 To obtain the last assertion of the theorem,  we  take the measures $e^{-i\xi}\mu_k$ instead of the measures $\mu_k$.  \bs
\medskip

{\bf Proof of Theorem \ref{T9}}.
Notice that   masses of the measures $\mu_k$ from \eqref{c2} are not assumed to be separated from zero, therefore we cannot make use of the result of Theorem \ref{T5}.

Let $\p(y)$ be nonnegative $C^\infty$-function with support in the unit ball such that \newline $\int\p(t)dt=1$. Pick $k\in(\N\cup\{0\})^d$, and set $\rho_\e=\e^d\p(\e y)\check\mu_k(y)$,
where $\check\mu_k$ is the inverse Fourier transform of the measure $\mu_k$. We have
\begin{equation}\label{j}
 \hat\rho_\e(s)=\int_{\R^d}\hat\p\left(\frac{s-x}{\e}\right)\mu_k(dx)=\left(\int_{|s-x|<1}+\int_{|s-x|\ge1}\right)\hat\p\left(\frac{s-x}{\e}\right)\mu_k(dx).
\end{equation}
Since $\hat\p\in S(\R^d)$, we get $\hat\p(y)=o(|y|^{-d-1})$ as $|y|\to\infty$ and $\hat\p((s-x)/\e)\to0$ for $s\neq x$ as $\e\to0$. Also, $\hat\p(0)=1$. Hence, by the Dominated Convergence Theorem,
$$
\lim_{\e\to0}\int_{|s-x|<1}\hat\p\left(\frac{s-x}{\e}\right)\mu_k(dx)=\begin{cases}\mu_k(\{\l\})=p_k(\l), & s=\l\in\L,\\  0, & s\not\in\L.\end{cases}
$$
Note that the masses $\mu_k(\{\l\})$ are uniformly bounded. Also,
$$
n_s(t):=\#(\L\cap B(s,t))=O(t^d)\qquad(t\to\infty).
$$
 Therefore,
$$
\left|\int\limits_{|s-x|\ge1}\hat\p\left(\frac{s-x}{\e}\right)\mu_k(dx)\right|\le C\int\limits_1^\infty(t/\e)^{-d-1}n_s(dt)\le\e^{d+1}(d+1)C\int\limits_1^\infty t^{-d-2}n_s(t)dt,
$$
and the left-hand side tends to zero as $\e\to0$.

Furthermore, by Theorem \ref{T11}, $\hat\mu_k$ has uniformly bounded masses and support of bounded density. Hence, $|\hat\mu_k|(B(0,r))=O(r^d)$ as $r\to\infty$ and the same is valid for the measures $\check\mu_k$.
Therefore,
$$
|\rho_\e|(B(0,r))\le\e^d|\check\mu_k|(B(0,1/\e)=O(1)\qquad (\e\to0).
$$
Since support of the measure $\rho_\e$ is bounded for each fixed $\e$, we see that the measures $\rho_\e$ form a bounded family of linear functionals on the space of continuous bounded functions on $\R^d$
 and, in particular, on its subspace $C(\hR)$. A closed  ball in the conjugate space $C^*(\hR)$ is a compact set in the weak-star topology. Hence there exists a measure $\hr_k$ on $\hR$ with a total variation
 $\|\hr_k\|<\infty$ such that for every $f\in C(\hR)$ there is a subsequence $\e'\to0$, for which we have $\la\rho_{\e'},f\ra\to\la\hr_k,f\ra$.
 Applying this to every character $x\in\R^d$ in the place of $f$, we obtain from \eqref{j}
 $$
 \hat\hr_k(x)=\lim_{\e'\to0}\hat\rho_{\e'}(x)=\begin{cases}p_k(\l), & x=\l\in\L,\\  0, & x\not\in\L.\end{cases}
 $$
 Note that $\hat\hr_k(x)$ is a continuous function with respect to the discrete topology on $\R^d$, and $|p_k(\l)|\le\|\hr_k\|$ for all $\l\in\R^d$.

Let $\psi$ be a $C^\infty$-function with compact support.  Since the inverse Fourier transform of the function $\psi(t-x)$ is $e^{2\pi i\la t,y\ra}\hat\psi(y)$, we get
\begin{equation}\label{f0}
(\psi\star\mu_k)(t)=\int_{\R^d}\psi(t-x)\mu_k(dx)=\int_{\R^d}e^{2\pi i\la y,t\ra}\hat\psi(y)\hat\mu_k(dy)=\sum_{\g\in\supp\hat\mu_k}\hat\mu_k(\g)\hat\psi(\g)e^{2\pi i\la\g,t\ra}.
\end{equation}
Clearly, $\hat\psi\in S(\R^d)$ and $\hat\psi(\g)=o(|\g|^{-d-1})$ as $\g\to\infty$. Also, the measure $\hat\mu_k$ has uniformly bounded masses and uniformly discrete support, therefore the series in the right-hand side is absolutely converges and $(\psi\star\mu_k)(t)\in W$. In particular, the measure $\mu_k$ is almost periodic.

 Furthermore, the function
$$
F(t)=\sum_{\|k\|\le K}(\psi\star\mu_k)(t)\overline{(\psi\star\mu_k)(t)}
$$
belongs to $W$ as well. Now suppose that $\psi$ has support in the ball $B(0,\eta(\L)/2)$ and $\psi(0)=1$. Taking into account that
\begin{equation}\label{p0}
(\psi\star\mu_k)(\l)=p_k(\l)\qquad\mbox{for}\quad \l\in\L,
\end{equation}
and using \eqref{c3}, we get
$$
 F(\l)=\sum_{\|k\|\le K} |p_k(\l)|^2\ge\a>0\qquad \forall\, \l\in\L.
$$
Using Proposition \ref{P3} with $h(z)=1/z$, we  construct the function
$$
g(t)=\sum_nc_ne^{2\pi i\la t,\tau_n\ra}\in W
$$
 with the property $g(\l)=1/F(\l)$ for $\l\in\L$.

Consider the dual pair $(\hR,\R^d_{discr})$. Set
$$
\ha=\sum_{\|k\|\le K}\hr_k(\t)\star\overline{\hr_k(-\t)},\quad \t\in\hR.
$$
 We get
 $$
 \hat\ha(\l)=F(\l)\quad \forall \l\in\L,\quad\mbox{and}\quad  \hat\ha(x)=0\quad\forall x\not\in\L.
 $$
Since  $\sum_n|c_n|<\infty$, we see that the series $\sum_nc_n\ha(y+\tau_n)$ converges with respect to the norm to a measure $\hb(y)$ such that
 $$
 \hat\hb(x)=\sum_n c_ne^{2\pi i\la x,\tau_n\ra}\hat\ha(x)=\begin{cases}1, & x\in\L,\\ 0, & \mbox{otherwise}. \end{cases}
 $$
By Theorem \ref{T6}, $\L$ belongs to the coset ring of $\R^d_{discr}$. By Theorem \ref{T7},
\begin{equation}\label{b2}
\L=\cup_{n=1}^N(A_n\setminus\cup_{s=1}^{S_n} B^s_n),
\end{equation}
where $A_n,\,B^s_n$ are cosets of discrete lattices.

Let $A,\,A'$  be two arbitrary cosets in $\R^d$ and $L,\,L'$ be the corresponding lattices. Clearly, $A\cap A'$ is a coset of $L\cap L'$. If $r(A\cap A')=d$, then the factor-groups $L/(L\cap L')$ and $L'/(L\cap L')$ are finite.
Therefore, $A\setminus A'=A\setminus(A\cap A')$ is a finite union of disjoint cosets of $L\cap L'$, and the same is valid for
$$
A\cup A'=(A\setminus(A\cap A'))\cup(A'\setminus(A\cap A'))\cup(A\cap A').
$$
We repeat this transformation for each pair of cosets with dimension $d$ of their intersection. After a finite number of steps, we get a representation similar to \eqref{b2} such that  $r(B^s_n)<d$
and $r(A_n\cap A_{n'})<d$ for $n\neq n'$.

Put $\d_A=\sum_{x\in A}\d_A$. Using the equalities $\d_{A\setminus A'}=\d_A-\d_{A\cap A'}$ and $\d_{A\cup A'}=\d_A+\d_{A'}-\d_{A\cap A'}$, we get
$$
   \d_\L=\sum_{j=1}^J\d_{D_j}+\sum_{j=1}^{J_1}\d_{E_j}-\sum_{j=1}^{J_2}\d_{F_j},
$$
where  $D_j,\,E_j,\,F_j$ are cosets such that $r(D_j)=d$ and $r(E_j)<d,\,r(F_j)<d$.

 Every coset $D_j$ is a translation of full-rank lattice,  hence $\d_{D_j}$ is $d$-periodic and $\sum_{j=1}^J\d_{D_j}$ is almost periodic.
Above we showed that all measures $\mu_k$ are almost periodic. By Proposition \ref{P2} ii), the measures $|\mu_k|$ are almost periodic, hence the measure $\sum_k|\mu_k|$
with support $\L$ is almost periodic as well.
 Using \eqref{c3} and Proposition \ref{P2} iii), we get that the measure $\d_\L$ is almost periodic. Therefore the left-hand side of the equality
 \begin{equation}\label{b}
 \sum_{j=1}^{J_1}\d_{E_j}- \sum_{j=1}^{J_2}\d_{F_j}=\d_\L-\sum_{j=1}^J\d_{D_j}
 \end{equation}
is an almost periodic measure as well. But its support  is contained in a finite union of hyperplanes and isn't relatively dense. Therefore the measure in the left-hand side of \eqref{b} is identically zero, and $\d_\L=\sum_{j=1}^J\d_{D_j}$.
Since  the measure $\d_\L$ has the unit masses at every point of $\L$, we see that the cosets $D_j$ are pairwise disjoint and  for some full-rank lattices $L_j$ and $\l_j\in\R^d$
\begin{equation}\label{m3}
\L=\cup_{j=1}^J D_j=\cup_{j=1}^J(\l_j+L_j).
\end{equation}

Recall that $\psi\star\mu_k\in W$. It follows from \eqref{p0} and \eqref{m3} that
\begin{equation}\label{m0}
 \mu_k=\sum_{j=1}^J (\psi\star\mu_k)\d_{L_j+\l_j}=\sum_{j=1}^J\sum_s a_{s,k}e^{2\pi i\la x,\g_{s,k}\ra}\d_{L_j+\l_j}
\end{equation}
for some $\g_{s,k}\in\R^d$ and $a_{s,k}\in\C$ such that $\sum_s|a_{s,k}|<\infty$.
  For every $\g\in\R^d$ there is $\b$ inside the parallelepiped generated by corresponding $L_j^*$ such that $\g-\b\in L_j^*$. Therefore,
$e^{2\pi i\la x,\g_{s,k}\ra}=e^{2\pi i\la x,\b_{s,k,j}\ra}$ for $x\in L_j$, and the set of all points $\b_{s,k,j}$ is bounded.
Therefore \eqref{m0} can be rewritten as
$$
\sum_{j=1}^J\sum_s\sum_{x\in L_j} b_s(k,j)e^{2\pi i\la x,\b_{s,k,j}\ra}\d_{x+\l_j}\quad\mbox{with}\quad b_s(k,j)= a_{s,k}e^{2\pi i\la \l_j,\g_{s,k}-\b_{s,k,j}\ra}.
$$
By \eqref{f2}, we obtain the statement of Theorem \ref{T9}.  \bs

\medskip

Here we actually proved the following statement:
\begin{Pro}\label{P4}
Let $\{\nu_n\}$ be a finite set of measures from $S^*(\R^d)$ such that for each $n$

a) $\supp\nu_n$ is uniformly discrete,

b) $\supp\hat\nu_n$ is locally finite of bounded density,

c) $\hat\nu_n$ is a measure with uniformly bounded masses.
\smallskip

\noindent Set $\hL=\cup_n\supp\nu_n$. If
$$
\inf_{\l\in\hL}\max_n|\nu_n(\l)|>0,
$$
then $\hL$ is  a  finite union of translates of several full-rank lattices.
\end{Pro}
Note that the boundedness of masses $\nu_n(\l)$ follows from \eqref{f0} with $\nu_k$ instead of $\mu_k$ and $\psi$ such that $\supp\psi\subset B(0,\eta(\supp\nu_n))$ and $\psi(0)=1$.

{\bf Remark 4}. It is not hard to check that conditions b) and c) can be replaced by

b') $\hat\nu_n$ is a purely point measure,

c')  $|\hat\nu_n|(B(0,r))=O(r^d)$ as $r\to\infty$.

\medskip
Our proof of Theorem \ref{T10} uses the following lemma
\begin{Le}\label{L2}
Let  measures
$$
\s_m=\sum_{\l\in\L_m} S_m(\l)\d_\l \quad\mbox{for}\quad m\in\N\cup\{0\},\,\|m\|\le M,
$$
with uniformly discrete supports $\L_m$ be almost periodic in $\R^d$.
If there are  constants $C$ and $h$  such that for all $\l\in\R^d$
\begin{equation}\label{p1}
p(\l)=\sum_{\|m\|\le M} \l^mS_m(\l)\quad\mbox{and}\quad |p(\l)|(1+|\l|)^{-h}<C,
\end{equation}
then $\s_m\equiv0$ for $\|m\|>h$.
 \end{Le}
Here we set $S_m(\l)=0$ for $\l\not\in\L_m$.
\smallskip

{\bf Proof of Lemma \ref{L2}}. Without loss of generality suppose  $M=\max\{\|m\|:\,\s_m\not\equiv0\}$.  Suppose the opposite $M>h$. Set
$$
A=\{m:\,\|m\|=M,\s_m\not\equiv0\},\quad N=\# A.
$$
 Let $\p$ be $C^\infty$-function such that $\p(0)=1,\ 0\le\p\le1,$ and $\supp\p\subset B(0,\rho)$ with $\rho<\eta(\supp\s_m)/2$. Set $\l\in\R^d$ such that
 $$
 \left|\sum_{m\in A}\l^mS_m(\l)\right|=\a\neq0
 $$
 (If the above sum  vanishes  for all $\l\in\R^d$, we can reduce the number $M$). Applying Proposition \ref{P2}, v) to the measures $\s_m$ with $\e<\frac{\a}{3N\max_{m\in A}(1+|\l|)^M}$, we find $R<\infty$
  such that every ball $B(s\l,R),\ s>0$, contains  a finite number of points $\{\l^{(j)}\}_{j=1}^J,\ J\le N$ with properties
 $$
     \forall m\in A\quad\exists j=\iota(m) \quad\mbox{such that}\quad |S_m(\l^{(j)})-S_m(\l)|<\e.
  $$
 Therefore, $|\l^m S_m(\l^{(j)})-\l^m S_m(\l)|<\a/3N$. Hence,
  $$
   \left|\sum_{m\in A}\l^mS_m(\l)-\sum_{j=1}^J\sum_{m:\iota(m)=j}\l^mS_m(\l^{(j)})\right|<\a/3, \quad\mbox{and}\quad\left|\sum_{j=1}^J\sum_{m:\iota(m)=j}\l^mS_m(\l^{(j)})\right|>2\a/3.
  $$
Here $J$ and $\l^{(j)}$ depend on $s$, but $R$ does not. Since $|\l^{(j)}-s\l|<R+\rho$, we see that $\l^{(j)}/s\to\l$ as $s\to\infty$, and $(\l^{(j)})^m s^{-M}\to\l^m$ for all $m\in A$. Therefore,  for $s$ large enough
 \begin{equation}\label{c1}
   \left|\sum_{j=1}^J\sum_{m:\iota(m)=j}(\l^{(j)})^ms^{-M}S_m(\l^{(j)})\right|>\a/3.
 \end{equation}
 On the other hand, $(\l^{(j)})^m s^{-M}\to0$ for $\|m\|<M$ and, by \eqref{p1}, $s^{-M}p(\l^{(j)})\to0$ as $s\to\infty$. Also, $S_m(\l)$ are uniformly bounded for all $m$ and $\l\in\R^d$.
  So we have
 $$
  \left|\sum_{m:\iota(m)=j}(\l^{(j)})^m s^{-M}S_m(\l^{(j)})\right|\le s^{-M}|p(\l^{(j)})|+\sum_{\|m\|<M}s^{-M}|(\l^{(j)})^mS_m(\l^{(j)})|\to0\quad (s\to\infty)
$$
uniformly with respect to $j$ and $\l^{(j)}$. Consequently,
$$
   \left|\sum_{j=1}^J\sum_{m:\iota(m)=j}(\l^{(j)})^ms^{-M}S_m(\l^{(j)})\right|\to0\quad (s\to\infty),
 $$
 which contradicts \eqref{c1}. \bs

\medskip
{\bf Proof of Theorem \ref{T10}}. By Theorem \ref{T11} b), $\hat f$ has the form
$$
\hat f(y)=\sum_k (2\pi iy)^k\hat\mu_k,\quad\mbox{where}\quad\hat\mu_k=\sum_{\|m\|\le H}\sum_{\g\in\G_k}q_{k,m}(\g)D^m\d_\g
$$
with uniformly bounded coefficients $q_{k,m}(\g)$ and uniformly discrete support $\G_k$, which satisfies \eqref{et}. Set
$$
  \nu_{k,m}=\sum_{\g\in\G_k}q_{k,m}(\g)\d_\g
$$
and apply Theorem \ref{T11} a) to every distribution
$$
\hat\mu_k=\sum_{\|m\|\le H}D^m\nu_{k,m},
$$
using the inverse Fourier transform instead of the direct one. We get
$$
  \check\nu_{k,m}=\sum_{\l\in\L_{k,m}}S_{k,m}(\l)\d_\l,
$$
with uniformly discrete $\L_{k,m}=\supp\check\nu_{k,m}$ and uniformly bounded $S_{k,m}(\l)$. Arguing as at the beginning of the proof of Theorem \ref{T11},
we obtain that the measures $\check\nu_{k,m}$ are almost periodic. Further,
$$
   \mu_k=\sum_{\|m\|\le M}(-2\pi i\l)^m\sum_{\l\in\L_{k,m}}S_{k,m}(\l)\d_\l,\quad\mbox{hence,}\quad p_k(\l)=\sum_{\|m\|\le M}(-2\pi i\l)^m S_{k,m}(\l).
$$
Using Lemma \ref{L2} with $\sigma_m=\check\nu_{k,m}$, we get $p_k(\l)=\sum_{\|m\|\le h(k)}(-2\pi i\l)^m S_{k,m}(\l)$. Therefore, for all $k$
$$
\frac{|p_k(\l)}{(1+|\l|)^{h(k)}}\le\sum_{\|m\|\le h(k)}\frac{(2\pi|\l|)^m}{(1+|\l|)^{h(k)}}\max_{\|m\|\le h(k)} |S_{k,m}(\l)|\le C\max_{\|m\|\le h(k)} |S_{k,m}(\l)|.
$$
Taking into account \eqref{c0}, we get the measures $\check\nu_{k,m}$ satisfies conditions of Proposition \ref{P4}. The statement of Theorem \ref{T10} follows from the embedding
 $\L\subset\cup_{k,m}\supp\check\nu_{k,m}$. \bs


\bigskip
I am very grateful to the referee for carefully reading my article and giving numerous remarks.


