# Generalized Fourier quasicrystals, almost periodic sets, and zeros of
Dirichlet series

Sergii Yu.Favorov  Sergii Favorov,
iii Karazin’s Kharkiv National University
iii Svobody sq., 4, Kharkiv, Ukraine 61022<EMAIL_ADDRESS>
> Abstract. Let $S(z)$ be an absolutely convergent Dirichlet series with a
> bounded spectrum and only real zeros $a_{n},\,n=1,2,\dots$, let $\mu$ be the
> sum of unit masses at points $a_{n}$. It is proven that the Fourier
> transform $\hat{\mu}$ in the sense of distributions is a purely point
> measure. Conversely, in terms of the properties of $\hat{\mu}$, a sufficient
> condition was found when $a_{n}$ are zeros of an absolutely convergent
> Dirichlet series with bounded spectrum. Also, in terms of the properties of
> $\hat{\mu}$, a criterion is established that $a_{n}$ are zeros of an almost
> periodic entire function of exponential growth. In all cases, the
> multiplicity of zeros is taken into account.
>
> Almost periodic sets, introduced by B.Levin and M.Krein in 1948, play
> important role in our investigations. In particular, we show a new simple
> representation of such sets.
>
> AMS Mathematics Subject Classification: 42A75, 42A38, 52C23
>
> Keywords: Fourier quasicrystal, Fourier transform in the sense of
> distributions, pure point measure, almost periodic entire function, almost
> periodic sets, zero set of entire functions

##  1\. Introduction

A crystalline measure on ${\mathbb{R}}^{d}$ is a complex measure with discrete
locally finite support, which is a temperate distribution and its
distributional Fourier transform $\hat{\mu}$ is also a measure with locally
finite support; if, in addition, the measures $|\mu|$ and $|\hat{\mu}|$ are
temperate distributions, then $\mu$ is called the Fourier quasicrystal.

The Fourier quasicrystal may be considered as a mathematical model for atomic
arrangement having a discrete diffraction pattern. There are a lot of papers
devoted to study properties of Fourier quasicrystals or, more generally,
crystalline measures. For example, one can mark collections of papers [2],
[17], in particular, the basic paper [12].

Measures of the form

(1) $\mu=\sum_{\lambda\in\Lambda}c_{\lambda}\delta_{\lambda},\qquad
c_{\lambda}\in{\mathbb{N}},$

are the most important case of Fourier quasicrystal. Recently A. Olevsky and
A. Ulanovsky [16], [15] showed that the supports $\Lambda$ of these measures
are zero sets of exponential polynomials with purely imaginary exponents and
only real zeros with multiplicities $c_{\lambda}$ at points
$\lambda\in\Lambda$ and, conversely, the zero sets of such exponential
polynomials are supports of some Fourier quasicrystals of the form (1).

In our article, using the same methods, we present analogues results for
measures (1) with the distribution Fourier transform

(2) $\hat{\mu}=\sum_{\gamma\in\Gamma}b_{\gamma}\delta_{\gamma},$

where $\Gamma$ is an arbitrary countable set. In this case the corresponding
Poisson’s formula

$\sum_{\lambda\in\Lambda}c_{\lambda}\hat{f}(\lambda)=\sum_{\gamma\in\Gamma}b_{\gamma}f(\gamma).$

also takes place for every function $f$ from Schwarz’s class. In order to
describe such measures, we use the concept of almost periodic sets, which was
introduced by M. Krein and B. Levin ([10], App. VI). In modern notations
(cf.[13], [18]), a locally finite set $\Lambda$ with multiplicities
$c_{\lambda}$ at points $\lambda\in\Lambda$ is almost periodic, if the
convolution of measure (1) with every continuous function with compact support
is an almost periodic function. We will write an almost periodic set like the
sequence $A=\\{a_{n}\\}_{n\in{\mathbb{Z}}}$, where each point $a_{n}=\lambda$
occurs $c_{\lambda}$ times. Therefore, almost periodic sets are in fact
multisets.

In Section 2 we give the original definition of almost periodic sets given by
Krein and Levin, which is equivalent to the one given above. Also we prove
some properties of almost periodic sets, in particular, show that such sets
have the form $\\{\alpha n+\phi(n)\\}_{n\in{\mathbb{N}}}$ with $\alpha>0$ and
an almost periodic mapping $\phi:\,{\mathbb{Z}}\to{\mathbb{R}}$.

In Section 3 we consider an absolutely convergent Dirichlet series with
bounded spectrum and a real set of zeros. We prove that the Fourier transform
of the measure $\mu_{A}$ corresponding to such a set $A$ is always a purely
point measure. Note that the zero set of any absolutely convergent Dirichlet
series (or, more general, any almost periodic holomorphic function and even
any holomorphic function with almost periodic modulus) is almost periodic
(cf.[8]).

In section 4 we investigate the inverse problem. We consider a measure $\mu$
of form (1) with the Fourier transform $\hat{\mu}$ of form (2) and find
sufficient conditions for which $A$ is the zero set of an absolutely
convergent Dirichlet series with bounded spectrum.

Here we assume that the measure $|\hat{\mu}|$ is a temperate distribution. By
[5], Lemma 1, the multiset $A=\\{a_{n}\\}_{n\in{\mathbb{Z}}}$, where each
point $a_{n}=\lambda$ occurs $c_{\lambda}$ times, is almost periodic. In [8]
it was proven that every almost periodic set $A\subset{\mathbb{R}}$ is exactly
the zero set of some entire almost periodic function. Every almost periodic
function is bounded on the real line, hence, by the Phragmen-Lindelöf
principle, any almost periodic entire function cannot grow less than
exponentially. We find a criterion for $A$ to be the zero set of an almost
periodic entire function of the exponential growth in terms of $\gamma$ and
$\beta_{\gamma}$ from (2).

##  2\. Almost periodic sets

######  Definition 1 (for example, see [1], [11]).

A continuous function $g(x)$ on the real line is almost periodic if for any
$\varepsilon>0$ the set of $\varepsilon$-almost periods

$E_{\varepsilon}=\\{\tau\in{\mathbb{R}}:\,\sup_{x\in{\mathbb{R}}}|g(x+\tau)-g(x)|<\varepsilon\\}$

is relatively dense, i.e., $E_{\varepsilon}\cap(x,x+L)\neq\emptyset$ for all
$x\in{\mathbb{R}}$ and some $L$ depending on $\varepsilon$.

For example every sum

$Q(x)=\sum q_{n}e^{2\pi ix\omega_{n}},\quad\omega_{n}\in{\mathbb{R}},\quad
q_{n}\in{\mathbb{C}},\qquad\sum_{n}|q_{n}|<\infty,$

is an almost periodic function.

Spectrum of an almost periodic function $g$ is the set

$\operatorname{sp}g=\left\\{w\in{\mathbb{R}}:\,\lim_{t\to\infty}\frac{1}{2T}\int_{-T}^{T}e^{-2\pi
i\omega x}g(x)dx\neq 0\right\\}.$

It is easy to see that $\operatorname{sp}Q(x)=\\{\omega_{n}:\,q_{n}\neq 0\\}$.
Note that spectra of almost periodic functions are at most countable.

######  Definition 2 (see [1], [11]).

A continuous function $g(z)$ on a strip

$S=\\{z=x+iy:\,-\infty\leq a<y<b\leq+\infty\\}\subset{\mathbb{C}}$

is almost periodic if for any $\alpha,\,\beta$ such that
$[\alpha,\beta]\subset(a,b)$ and $\varepsilon>0$ the set of
$\varepsilon$-almost periods

$E_{\alpha,\beta,\varepsilon}=\\{\tau\in{\mathbb{R}}:\,\sup_{x\in{\mathbb{R}},\alpha\leq
y\leq\beta}|g(x+\tau+iy)-g(x+iy)|<\varepsilon\\}$

is relatively dense, i.e.,
$E_{\alpha,\beta,\varepsilon}\cap(x,x+L)\neq\emptyset$ for all
$x\in{\mathbb{R}}$ and some $L$ depending on $\varepsilon,\alpha,\beta$.

Just as it was done in [10], we could define an almost periodic set in a
strip. Here we give this definition only in the simplified form. The general
definition, which takes into account the behavior of $A$ near the boundary,
was given by H.Tornehave in [19]. The connection between almost periodic sets
of general form and zeros of holomorphic almost periodic functions in terms of
Chern cohomology see [3].

######  Definition 3 (M. Krein and B. Levin [10], App. VI).

Let $S$ be a horizontal strip of a finite width. A discrete locally finite
multiset $A=\\{a_{n}\\}_{n\in{\mathbb{Z}}}\subset S$ is almost periodic if for
any $\varepsilon>0$ the set of its $\varepsilon$-almost periods

(3) $E_{\varepsilon}=\\{\tau\in{\mathbb{R}}:\,\exists\ \text{ a bijection
}\sigma:{\mathbb{Z}}\to{\mathbb{Z}}\quad\text{such that}\ \sup_{n}|a_{n}+\tau-
a_{\sigma(n)}|<\varepsilon\\}$

has nonempty intersection with every interval $(x,x+L_{\varepsilon})$.

In our article we consider, as a rule, only the case of almost periodic sets
on the real line.

Set $\mu_{A}=\sum_{n}\delta_{a_{n}}$. Clearly, the mass of $\mu_{A}$ in any
point $x\in{\mathbb{R}}$ is equal to the multiplicity of this point in the
sequence $\\{a_{n}\\}_{n\in{\mathbb{Z}}}$.

It was proved in [8] that almost periodicity of $A$ is equivalent to almost
periodicity of convolution $\mu_{A}\star\varphi$ for every
$C^{\infty}$-function $\varphi(x),\,x\in{\mathbb{R}}$, with compact support.
But it is easy to replace $C^{\infty}$-functions by continuous functions with
compact support.

Indeed, take $C^{\infty}$-function $\varphi\geq 0$ such that $\varphi(x)\equiv
1$ for $0<x<1$. If $\mu_{A}\star\varphi$ is almost periodic, then it is
uniformly bounded, hence $\mu_{A}[x,x+1]<k_{1}$ for all $x\in{\mathbb{R}}$
with some constant $k_{1}$. For any continuous function $\psi$ with support in
$(0,1)$ one can take $\varphi\in C^{\infty}$ such that
$\sup_{x\in{\mathbb{R}}}|\psi(x)-\varphi(x)|<\varepsilon/k_{1}$. We obtain
that every $\varepsilon$-almost period of $\mu_{A}\star\varphi$ is
$2\varepsilon$-almost period of $\mu_{A}\star\psi$.

By the way, we gave the proof of the following proposition

######  Proposition 1 ([8]).

For any almost periodic set there is $k_{1}\in{\mathbb{N}}$ such that
$\\#A\cap[x,x+1]\leq k_{1}$. Also, $\\#A\cap[x,x+h)\leq k_{1}(h+1)$.

Here and below, $\\#H$ means the number of points in the multiset $H$ taking
into account their multiplicities.

######  Proposition 2.

For any almost periodic set there is $k_{2}\in{\mathbb{N}}$ such that for
every $h>0$ and every half-intervals $[x_{1},x_{1}+h),\,[x_{2},x_{2}+h)$ we
have $|\\#A\cap[x_{1},x_{1}+h)-\\#A\cap[x_{2},x_{2}+h)|\leq k_{2}$. Also, for
every $x\in{\mathbb{R}},\,h>0,\,M\in{\mathbb{N}}$

$|\\#A\cap[x,x+h)-(1/M)\\#A\cap[x,x+Mh)|\leq k_{2}.$

Proof. Let $L_{1},\,E_{1}$ be defined in (3), and $\tau\in
E_{1}\cap[x_{1}-x_{2},L_{1}+x_{1}-x_{2})$. Since
$[x_{2},x_{2}+h)+\tau\subset[x_{1},x_{1}+L_{1}+h)$, we see that to each
$a_{n}\in[x_{2},x_{2}+h)$ assign a point
$a_{\sigma(n)}\in[x_{1}-1,x_{1}+L_{1}+h+1)$. Therefore,

$\\#A\cap[x_{2},x_{2}+h)\leq\\#A\cap[x_{1},x_{1}+h)+\\#A\cap[x_{1}-1,x_{1})+\\#A\cap[x_{1}+h,x_{1}+h+L_{1}+1).$

By Proposition 1, the last two terms are bounded by $k_{1}+(L_{1}+2)k_{1}$.
The proof of the opposite inequality is the same.

To prove the second assertion, we have to add together all the inequalities

$\\#A\cap[x,x+h)-k_{2}\leq\\#A\cap[x+(m-1)h,x+mh)\leq
k_{2}+\\#A\cap[x,x+h),\quad m=1,2,\dots,M.$

######  Proposition 3.

Let $A$ be an almost periodic set. There is a strictly positive density $d$
such that for any $\eta>0$ and any half-interval $I$ with length
$l(I)>N_{\eta}$ we have

$\left|\frac{\\#A\cap I}{l(I)}-d\right|<\eta.$

This result was generalized to multidimensional Euclidean spaces in [7].

Proof of Proposition 3. Let
$I_{1}=[x_{1},x_{1}+h_{1}),I_{2}=[x_{2},x_{2}+h_{2})$ be two half-intervals
such that $h_{1}/h_{2}=p/q,\,p,q\in{\mathbb{N}}$. We have

$\frac{\\#A\cap I_{1}}{h_{1}}-\frac{\\#A\cap I_{2}}{h_{2}}=\frac{\\#A\cap
I_{1}}{h_{1}}-\frac{\\#A\cap qI_{1}}{qh_{1}}+\frac{\\#A\cap
qI_{1}}{qh_{1}}-\frac{\\#A\cap pI_{2}}{ph_{2}}+\frac{\\#A\cap
pI_{2}}{ph_{2}}-\frac{\\#A\cap I_{2}}{h_{2}}.$

Applying Proposition 2, we get

(4) $\left|\frac{\\#A\cap I_{1}}{h_{1}}-\frac{\\#A\cap
I_{2}}{h_{2}}\right|\leq\frac{k_{2}}{h_{1}}+\frac{k_{2}}{qh_{1}}+\frac{k_{2}}{h_{2}}\leq
k_{2}\left(\frac{2}{h_{1}}+\frac{1}{h_{2}}\right).$

For arbitrary $h_{1},\,h_{2}$ take a half-interval
$I^{\prime}=[x_{1},x_{1}+h^{\prime})$ such that $h_{1}<h^{\prime}<h_{1}+1$ and
$h^{\prime}/h_{2}$ rational. We have

$\left|\frac{\\#A\cap I_{1}}{h_{1}}-\frac{\\#A\cap
I^{\prime}}{h^{\prime}}\right|\leq\frac{\\#A\cap[x_{1}+h_{1},x_{1}+h^{\prime})}{h^{\prime}}+\frac{\\#A\cap
I_{1}}{h_{1}h^{\prime}}.$

By Proposition 1, we obtain

$\left|\frac{\\#A\cap I_{1}}{h_{1}}-\frac{\\#A\cap
I^{\prime}}{h^{\prime}}\right|\leq\frac{k_{1}}{h^{\prime}}+\frac{k_{1}(h_{1}+1)}{h_{1}h^{\prime}}.$

Applying (4) with $I^{\prime}$ instead of $I_{1}$, we obtain for all
$I_{1},\,I_{2}$

$\left|\frac{\\#A\cap I_{1}}{l(I_{1})}-\frac{\\#A\cap
I_{2}}{l(I_{2})}\right|\leq
k_{2}\left(\frac{2}{l(I_{1})}+\frac{1}{l(I_{2})}\right)+k_{1}\left(\frac{2}{l(I_{1})}+\frac{1}{l(I_{1})^{2}}\right).$

Therefore there is a limit

$d=\lim_{l(I)\to\infty}\frac{\\#A\cap I}{l(I)}.$

It is easy to check that the set $A$ has nonempty intersection with every
interval of length $2+L_{1}$, hence this limit is strictly positive.

######  Theorem 1.

Let $A=\\{a_{n}\\}\subset{\mathbb{R}}$ be an almost periodic set of density
$d$ such that $a_{n}\leq a_{n+1}$ for all $n\in{\mathbb{Z}}$. Then

(5) $a_{n}=n/d+\phi(n)\quad\text{with an almost periodic
mapping}\quad\phi:\,Z\to{\mathbb{R}}.$

Remark 1. The incomplete proof of this Theorem was given in [6].

Remark 2. The converse assertion is simple, because for every
$\varepsilon$-almost period $\tau\in{\mathbb{Z}}$ of the mapping $\phi(n)$ the
number $\tau/d$ is $\varepsilon$-almost period of an almost periodic set
$A=\\{n/d+\phi(n)\\}$ with any $d>0$ and $\sigma(n)=n+\tau$.

Proof of Theorem 1. We may suppose that $a_{0}<a_{1}$. It follows from
Proposition 1 that every interval of length $1$ contains at least one
subinterval of length $1/(2k_{1})$ that does not intersect $A$. Take
$\varepsilon<\min\\{1/(6k_{1}),(a_{1}-a_{0})/3\\}$. Divide ${\mathbb{R}}$ into
an infinite number of disjoint half-intervals
$I_{j}=(t_{j},t_{j+1}],\,j\in{\mathbb{Z}}$ such that $t_{j+1}-t_{j}<2$ and
$A\cap(t_{j}-2\varepsilon,t_{j}+2\varepsilon)=\emptyset$ for all $j$.

Let $\tau$ be any number from $E_{\varepsilon}$ in (3), and let $\sigma$ be
the corresponding bijection. Then $\rho(j)\in{\mathbb{Z}}$ corresponds to any
$j$ such that $\sigma$ is the bijection of $A\cap I_{j}$ to $A\cap
I_{\rho(j)}$. Hence, $\\#(A\cap I_{j})=\\#(A\cap I_{\rho(j)})$. Let
$\sigma_{j}$ be the monotone increasing bijection of $A\cap I_{j}$ on $A\cap
I_{\rho(j)}$. Check that

(6) $|a_{n}+\tau-a_{\sigma_{j}(n)}|<\varepsilon\qquad\forall\,a_{n}\in I_{j}.$

Suppose the contrary. Let $n_{0}$ be the minimal number such that (6) does not
satisfy. If $a_{n_{0}}+\tau+\varepsilon\leq a_{\sigma_{j}(n_{0})}$, then
$a_{n}+\tau+\varepsilon\leq a_{k}$ for all $n\leq n_{0}$ and
$k\geq\sigma_{j}(n_{0})$, $a_{n}\in I_{j},\,a_{k}\in I_{\rho(j)}$. Therefore,
$k\neq\sigma(n)$ for these numbers, and $\sigma$ gives a correspondence
between points from the set $\\{n\leq n_{0}:\,a_{n}\in I_{j}\\}$ and points
from the set $\\{k<\sigma_{j}(n_{0}):\,a_{k}\in I_{\rho(j)}\\}$. But by
definition of $\sigma_{j}$, we have

$\\#\\{n\leq n_{0}:\,a_{n}\in I_{j}\\}=\\#\\{k\leq\sigma_{j}(n_{0}):\,a_{k}\in
I_{\rho(j)}\\}=1+\\#\\{k<\sigma_{j}(n_{0}):\,a_{k}\in I_{\rho(j)}\\}.$

We get a contradiction.

If $a_{n_{0}}+\tau\geq a_{\sigma_{j}(n_{0})}+\varepsilon$, then
$a_{n}+\tau\geq a_{k}+\varepsilon$ for all $n\geq n_{0}$ and
$k\leq\sigma_{j}(n_{0})$, $a_{n}\in I_{j},\,a_{k}\in I_{\rho(j)}$. Therefore,
$k\neq\sigma(n)$ for these numbers, and $\sigma$ gives a correspondence
between points from the set $\\{n\geq n_{0}:\,a_{n}\in I_{j}\\}$ and points
from the set $\\{k>\sigma_{j}(n_{0}):\,a_{k}\in I_{\rho(j)}\\}$. But by
definition of $\sigma_{j}$, we have

$\\#\\{n\geq n_{0}:\,a_{n}\in I_{j}\\}=\\#\\{k\geq\sigma_{j}(n_{0}):\,a_{k}\in
I_{\rho(j)}\\}=1+\\#\\{k>\sigma_{j}(n_{0}):\,a_{k}\in I_{\rho(j)}\\}.$

We get a contradiction as well.

Since numbers $\\#(A\cap I_{j})$ and $\\#(A\cap I_{\rho(j)})$ coincide, we see
that the differences between indices of the first elements in these sets
coincide for all $j$. Hence there is a number $h\in{\mathbb{Z}}$ so that
inequality (3) satisfies for all $n\in{\mathbb{N}}$ with $\sigma(n)=n+h$.

It follows from the definition of $\tau$ for all $k\in{\mathbb{N}}$ and
$N\in{\mathbb{N}}$

$-\varepsilon<a_{kh}-\tau-
a_{(k-1)h}<\varepsilon,\quad\text{and}\quad-N\varepsilon<a_{Nh}-N\tau-
a_{0}<N\varepsilon.$

Let $I$ be the smallest half-interval containing $a_{0}$ and $a_{Nh}$. The
last inequality implies that its length satisfies the inequality

$N\tau-N\varepsilon<l(I)<N\tau+N\varepsilon.$

On the other hand, taking into account that ends of $I$ may be points of $A$
with multiplicity at most $k_{1}$, we have

$Nh-2(k_{1}-1)\leq\\#A\cap I\leq Nh+2(k_{1}-1).$

Therefore,

$\frac{Nh-2(k_{1}-1)}{N\tau+N\varepsilon}\leq\frac{\\#A\cap
I}{l(I)}\leq\frac{Nh+2(k_{1}-1)}{N\tau-N\varepsilon}.$

Passing to the limit as $N\to\infty$ and using Proposition 3, we obtain the
inequality

$\tau-\varepsilon\leq h/d\leq\tau+\varepsilon.$

Set $\phi(n):=a_{n}-n/d$. We get for all $n\in{\mathbb{Z}}$

$\phi(n+h)-\phi(n)=a_{n+h}-(n+h)/d-a_{n}+n/d=a_{\sigma(n)}-(a_{n}+\tau)+(\tau-h/d).$

Using (3), we obtain $|\phi(n+h)-\phi(n)|<2\varepsilon$. Therefore, $h$ is
$2\varepsilon$-almost period of the function $\phi$. The set of
$\varepsilon$-almost periods $\tau$ of $A$ is relatively dense, therefore the
set of such integers $h$ is relatively dense as well.

######  Corollary 1.

For any almost periodic set $A=\\{a_{n}\\}$ such that $0\not\in A$ there is a
finite limit

$\lim_{N\to\infty}\sum_{|a_{n}|<N}1/a_{n}.$

Moreover, the sum

$\frac{1}{z-a_{0}}+\sum_{n\in{\mathbb{N}}\setminus\\{0\\}}\left[\frac{1}{z-a_{n}}+\frac{1}{z-a_{-n}}\right].$

converges absolutely and uniformly on every disjoint with $A$ compact set $K$.

Proof. Let $A=\\{n/d+\phi(n)\\}_{n\in{\mathbb{Z}}}$. Since the numbers
$\phi(n)$ are uniformly bounded, we see that the sums

$\sum_{n\in{\mathbb{Z}},|a_{n}|<N}\frac{1}{a_{n}}\quad\mbox{and}\quad\sum_{n\in{\mathbb{Z}},|n|<dN}\frac{1}{n/d+\phi(n)}$

differ for a uniformly bounded with respect to $N$ number of terms, and each
of these terms tends to $0$ as $N\to\infty$. Then

$\sum_{n\in{\mathbb{Z}},0<|n|<N}\frac{1}{n/d+\phi(n)}=\sum_{n\in{\mathbb{N}},0<n<N}\frac{\phi(n)+\phi(-n)}{\phi(n)\phi(-n)+n\phi(-n)/d-n\phi(n)/d-(n/d)^{2}}.$

The first assertion follows from Cauchy criterium. The second one follows from
the absolutely convergence of the series

$\sum_{n\in{\mathbb{N}}\setminus\\{0\\}}\left[\frac{1}{z-a_{n}}+\frac{1}{z-a_{-n}}\right]=\sum_{n\in{\mathbb{N}}\setminus\\{0\\}}\left[\frac{2z-\phi(-n)-\phi(n)}{(n/d+\phi(n)-z)(-n/d+\phi(-n)-z)}\right].\phantom{XXX}\leavevmode\nobreak\
\hfill\rule{7.0pt}{7.0pt}$

In [10], App. VI, M.Krein and B.Levin considered zero sets $Z_{f}$ of entire
almost periodic functions $f$ of exponential growth. They proved that if
$Z_{f}\subset{\mathbb{R}}$, then its zeros $a_{n}$ form an almost periodic
set, which satisfy (5) and

(7)
$\sup_{\tau\in{\mathbb{Z}}}\sum_{n\in{\mathbb{Z}}\setminus\\{0\\}}n^{-1}[\phi(n+\tau)-\phi(n)]<\infty.$

On the other hand, they proved that any almost periodic set
$A\subset{\mathbb{R}}$ satisfying conditions (5) and (7) is the set of zeros
of an entire almost periodic function of exponential growth.

It follows from Theorem 1 that condition (5) can be omitted in the last
result.

Theorem 1 was generalized by W.Lawton [9] to almost periodic sets in
${\mathbb{R}}^{m},\,m>1,$ whose spectrum is contained in a finitely generated
additive group.

##  3\. Zeros of infinite exponential sums

Denote by $S({\mathbb{R}})$ the Schwartz space of test functions $\varphi\in
C^{\infty}({\mathbb{R}})$ with the finite norms

$N_{n,m}(\varphi)=\sup_{{\mathbb{R}}}\max_{k\leq
m}|(1+|x|^{n})\varphi^{(k)}(x)|,\quad n,m=0,1,2,\dots$

These norms generate the topology on $S({\mathbb{R}})$. Elements of the space
$S^{*}({\mathbb{R}})$ of continuous linear functionals on $S({\mathbb{R}})$
are called temperate distributions.

The Fourier transform of a temperate distribution $f$ is defined by the
equality

$\hat{f}(\varphi)=f(\hat{\varphi})\quad\mbox{for all}\quad\varphi\in
S({\mathbb{R}}),$

where

$\hat{\varphi}(t)=\int_{{\mathbb{R}}^{d}}\varphi(x)e^{-2\pi ixt}dx$

is the Fourier transform of the function $\varphi$. By $\check{\varphi}$ we
denote the inverse Fourier transform of $\varphi$. The Fourier transform is
the bijection of $S({\mathbb{R}})$ on itself and the bijection of
$S^{*}({\mathbb{R}})$ on itself.

Let $\mathfrak{T}$ be the class of exponential sums

$f(x)=\sum_{n}q_{n}e^{2\pi i\omega_{n}x},\quad
q_{j}\in{\mathbb{C}}\setminus\\{0\\}$

with finite Wiener’s norm $\|f\|_{W}:=\sum_{n}|q_{n}|$ and a bounded spectrum
$\Omega:=\\{\omega_{n}\\}\subset{\mathbb{R}}$.

Every function $f(x)\in\mathfrak{T}$ expands to an entire almost periodic
function $f(z)$ of exponential type $\sigma=\max_{n}|\omega_{n}|$; the zero
set $A=\\{a_{n}\\}$ of $f(z)$ lies in some horizontal strip of a finite width
if and only if $\inf\Omega\in\Omega$ and $\sup\Omega\in\Omega$ (cf.[10],
Ch.VI, Cor.2). Moreover, $A$ is an almost periodic set (cf.[10], App.VI, Lemma
1).

If $0\not\in A$ and $a_{n}=\alpha n+\phi(n)$ with almost periodic
$\phi:\,{\mathbb{Z}}\to{\mathbb{R}}$ such that the Fourier series of $\phi$
converges absolutely, then the function

$(1-z/a_{0})\prod_{n\in{\mathbb{N}}}(1-z/a_{n})(1-z/a_{-n})$

expands in an absolutely convergent exponential series (cf.[10], App. VI,
Th.9), hence it belongs to $\mathfrak{T}$.

######  Theorem 2.

Suppose that $f\in\mathfrak{T}$ has the zero set
$A=\\{a_{n}\\}\subset{\mathbb{R}}$, and $\mu_{A}=\sum_{n}\delta_{a_{n}}$. Then
the Fourier transform $\hat{\mu}_{A}$ is a pure point measure.

Remark. Y.Meyer ([14], Th.5.16) proved that under condition $a_{n}=\alpha
n+\phi(n)$ with an absolutely convergent Fourier series for $\phi$ the
temperate distribution $\hat{\mu}_{A}$ is a pure point measure. It follows
from [10], App. VI, Th.9, that the above Theorem 2 is a strengthening of
Meyer’s result.

Proof. It follows from Proposition 1 that the measure $\mu_{A}$ satisfies the
condition $\mu_{A}([-r,r])=O(r)$ as $r\to\infty$. Hence the measure $\mu_{A}$
and the distribution $\hat{\mu}_{A}$ are temperate distributions. In order to
prove that $\hat{\mu}_{A}$ is a measure, we will check the estimate

(8) $|(\hat{\mu}_{A},\varphi)|\leq C\max_{|t|<T}|\varphi(t)|$

for any $T<\infty$ and any $C^{\infty}$-function $\varphi$ with support on the
interval $(-T,T)$. If this is the case, the distribution $\hat{\mu}$ has a
unique expansion to a linear functional on the space of continuous functions
$g$ on $[-T,T]$ such that $g(\alpha)=g(\beta)=0$ with bound (8). Since we can
expand this functional to the space of all continuous functions on $[-T,T]$
with bound (8) too, we see that $\hat{\mu}_{A}$ is a complex measure.

Let $\varphi$ be $C^{\infty}$-function with support in $(-T,T)$. Set

$\Phi(z)=\int_{-\infty}^{\infty}\varphi(t)e^{-2\pi itz}dt.$

Clearly, $\Phi(z)$ is an entire function, which equals the Fourier transform
of the function $\varphi(t)e^{2\pi ty}$. Therefore, $\Phi(x+iy)$ belongs to
$S({\mathbb{R}})$ for each fixed $y\in{\mathbb{R}}$, and we have for its
converse Fourier transform

(9) $\check{\Phi}(\omega+iy)=\varphi(\omega)e^{2\pi\omega
y},\qquad\omega\in{\mathbb{R}}.$

Then $f(z)$ is almost periodic, hence Lemma 1 from [10], Ch.6, implies that
for every $\varepsilon>0$ and $s<\infty$ there is $m=m(\varepsilon,s)>0$ such
that

$|f(z)|\geq m\quad\mbox{for}\quad|\mbox{Im }z|\leq s\quad\mbox{and}\quad
z\not\in A(\varepsilon):=\\{z:\,\operatorname{dist}(z,A)<\varepsilon\\}.$

By Proposition 1, for $\varepsilon$ small enough each connected component of
$A(\varepsilon)$ contains no segment of length $1$, hence its diameter is less
than $1$. Therefore there are two sequences
$R_{k}\to+\infty,\,R^{\prime}_{k}\to-\infty$ such that

$|f(x+iy)|>m\quad\text{for}\quad x=R_{k}\quad\text{or}\quad
x=R^{\prime}_{k},\quad|y|<1.$

Consider integrals of the function $\Phi(z)f^{\prime}(z)f^{-1}(z)$ over the
boundaries of the rectangles
$\Pi_{k}=\\{z=x+iy:\,R^{\prime}_{k}<x<R_{k},-s^{\prime}<y<s\\}$, where the
numbers $s,\,s^{\prime}>0$ will be chosen later. Since $\Phi(x\pm iy)$ tends
to zero as $x=R_{k}\to+\infty,\,x=R^{\prime}_{k}\to-\infty$ uniformly with
respect to $-s^{\prime}\leq y\leq s$, we obtain that these integrals tend to

(10)
$\int_{+\infty}^{-\infty}\Phi(x+is)f^{\prime}(x+is)f^{-1}(x+is)dx+\int_{-\infty}^{+\infty}\Phi(x-is^{\prime})f^{\prime}(x-is^{\prime})f^{-1}(x-is^{\prime})dx=:I_{1}+I_{2}..$

By the Theorem on residues

(11) $I_{1}+I_{2}=2\pi
i\sum_{\lambda:f(\lambda)=0}\operatorname{Res}_{\lambda}\Phi(z)f^{\prime}(z)f^{-1}(z)=2\pi
i\sum_{\lambda:f(\lambda)=0}a(\lambda)\Phi(\lambda)=2\pi i(\mu_{A},\Phi),$

where $a(\lambda)$ is the multiplicity of $f(z)$ at the point $\lambda$.

Set $\omega_{1}=\inf\Omega$. Then the corresponding coefficient $q_{1}$ does
not vanish. Taking into account that $\sum_{n}|q_{n}|<\infty$, we can take
numbers first $M$ and then $s$ such that

(12)
$\sum_{n>M}|q_{n}/q_{1}|<1/3,\phantom{XXXXXXXX}e^{2\pi(\omega_{1}-\omega^{\prime})s}\sum_{n=2}^{\infty}|q_{n}/q_{1}|<1/3,$

where $\omega^{\prime}=\min_{2\leq n\leq M}\omega_{n}$. We have for $z=x+is$

$f(z)=q_{1}e^{2\pi
i(x+is)\omega_{1}}\left(1+\sum_{n=2}^{M}(q_{n}/q_{1})e^{2\pi
i(\omega_{n}-\omega_{1})(x+is)}+\sum_{n>M}(q_{n}/q_{1})e^{2\pi
i(\omega_{n}-\omega_{1})(x+is)}\right).$

Set

$H(x):=\sum_{n=2}^{\infty}(q_{n}/q_{1})e^{2\pi
i(\omega_{n}-\omega_{1})x}e^{2\pi(\omega_{1}-\omega_{n})s}=\sum_{n=2}^{\infty}h_{n}(s)e^{2\pi
i(\omega_{n}-\omega_{1})x}.$

By (12), we have $\|H\|_{W}<2/3$. Since $\|\cdot\|_{W}$ is the norm on the
algebra of all absolutely convergent exponential sums, we get

(13)
$(1+H(x))^{-1}=\sum_{j=0}^{\infty}(-1)^{j}H^{j}(x),\qquad\|(1+H)^{-1}\|_{W}\leq\sum_{j=0}^{\infty}\|H^{j}\|_{W}<3.$

We have

$f^{-1}(x+is)=q_{1}^{-1}e^{2\pi\omega_{1}s}e^{-2\pi
i\omega_{1}x}(1+H(x))^{-1},\qquad f^{\prime}(x+is)=\sum_{n=1}^{\infty}2\pi
i\omega_{n}q_{n}e^{-2\pi\omega_{n}s}e^{2\pi i\omega_{n}x},$

and

(14) $f^{\prime}(x+is)f^{-1}(x+is)=\sum_{n=1}^{\infty}2\pi
i\omega_{n}(q_{n}/q_{1})e^{-2\pi(\omega_{n}-\omega_{1})s}e^{2\pi
i(\omega_{n}-\omega_{1})x}(1+H(x))^{-1}.$

Rewrite $f^{\prime}f^{-1}$ in the form

$f^{\prime}(x+is)f^{-1}(x+is)=\sum_{\gamma\in\Gamma_{1}}p_{\gamma}e^{2\pi
i\gamma x},\quad p_{\gamma}=p_{\gamma}(s)\in{\mathbb{C}},$

with some countable $\Gamma_{1}\subset{\mathbb{R}}_{+}\cup\\{0\\}$. Since
$\Omega$ is bounded, we obtain from (13) and (14)

$\sum_{\gamma\in\Gamma_{1}}|p_{\gamma}|=\|f^{-1}(x+is)f^{\prime}(x+is)\|_{W}\leq
6\pi\max_{n}\\{|\omega_{n}|e^{2\pi(\omega_{n}-\omega_{1})s}\\}\sum_{n}|q_{n}/q_{1}|=:C_{f}.$

The function $\Phi(x+is)$ belongs to $S({\mathbb{R}})$ for $s$ fixed, hence
$|x|^{2}\Phi(x+is)\to 0$ as $|x|\to\infty$. Changing the order of integration
and summation and taking into account (9), we obtain for the first integral in
(10)

$I_{1}=-\sum_{\gamma\in\Gamma_{1}}p_{\gamma}\int_{-\infty}^{+\infty}\Phi(x+is)e^{2\pi
i\gamma x}dx=-\sum_{\gamma\in\Gamma_{1}}p_{\gamma}e^{2\pi\gamma
s}\varphi(\gamma).$

Since $\operatorname{supp}\varphi\subset(-T,T)$, we get the bound

$|I_{1}|\leq C_{f}e^{2\pi Ts}\max_{|t|\leq T}|\varphi(t)|.$

The similar arguments show that the second integral in (10) with the
appropriate $s^{\prime}$ has the same bound.

Since $\hat{\varphi}(x)=\Phi(x)$, we get from (11)

$(\mu_{A},\hat{\varphi})=(2\pi i)^{-1}(I_{1}+I_{2}).$

Therefore,

$|(\hat{\mu}_{A},\varphi)|=|(\mu_{A},\hat{\varphi})|\leq C(f,T)\sup_{|y|\leq
T}|\varphi(y)|,$

and $\hat{\mu}_{A}$ is a measure. Since $\mu_{A}$ is almost periodic, Theorem
5.5 from [14] implies that $\hat{\mu}_{A}$ is a pure point measure.

##  4\. Entire functions with the given almost periodic zero sets

In this section we assume that a measure $\mu$ of form (1) is a temperate
distribution, its Fourier transform $\hat{\mu}$ is a pure point measure of
form (2), the measure $|\hat{\mu}|$ is a temperate distribution, and
$A=\\{a_{n}\\}_{n\in{\mathbb{Z}}}$ is a multiset, where each point
$a_{n}=\lambda\in\operatorname{supp}\mu$ occurs $c_{\lambda}$ times. By [5],
Lemma 1, $\mu$ is an almost periodic measure and $A$ is an almost periodic
set. Since $\hat{\mu}$ is also a measure, Theorem 5.5 from [14] implies that
every number $b_{\gamma}$ from (2) equals the corresponding Fourier
coefficient of the measure $\mu$, i.e.,

$b_{\gamma}=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}e^{-2\pi i\gamma
x}\mu(dx).$

In particular, $b_{-\gamma}=\bar{b}_{\gamma}$ and $b_{0}\geq 0$. Moreover,
$b_{0}$ coincides with the density $d$ of the set $A$. Also, it can be checked
(cf.[4]) that the condition $|\hat{\mu}|\in S^{*}({\mathbb{R}})$ implies

(15)
$|\hat{\mu}|(-r,r)=\sum_{|\gamma|<r}|b_{\gamma}|=O(r^{\kappa})\quad\text{as}\quad
r\to\infty$

with some $\kappa<\infty$.

In what follows we will suppose that $0\not\in A$. By Corollary 1, the set $A$
satisfies Lindelof’s condition. Hence, the function

(16) $f(z)=(1-z/a_{0})\prod_{n\in{\mathbb{N}}}(1-z/a_{n})(1-z/a_{-n})$

is an entire function of exponential type with the zero set $A$. Note that
$f(\bar{z})=\overline{f(z)}$. Also, introduce the notations

${\mathbb{R}}_{+}:=\\{x\in{\mathbb{R}}:x>0\\},\,{\mathbb{R}}_{-}:=-{\mathbb{R}}_{+},\,{\mathbb{C}}_{+}:=\\{z\in{\mathbb{C}}:\mbox{Im
}z>0\\},\,{\mathbb{C}}_{-}:=-{\mathbb{C}}_{+}.$

######  Proposition 4.

For all $z=x+iy\in{\mathbb{C}}_{+}$

(17)
$\frac{f^{\prime}(z)}{f(z)}=\frac{1}{z-a_{0}}+\sum_{n\in{\mathbb{N}}}\left[\frac{1}{z-a_{n}}+\frac{1}{z-a_{-n}}\right]=-2\pi
i\sum_{\gamma\in\Gamma\cap{\mathbb{R}}_{+}}b_{\gamma}e^{2\pi i\gamma z}-\pi
id,$

and for all $z=x+iy\in{\mathbb{C}}_{-}$

(18)
$\frac{f^{\prime}(z)}{f(z)}=\frac{1}{z-a_{0}}+\sum_{n\in{\mathbb{N}}}\left[\frac{1}{z-a_{n}}+\frac{1}{z-a_{-n}}\right]=2\pi
i\sum_{\gamma\in\Gamma\cap{\mathbb{R}}_{-}}b_{\gamma}e^{2\pi i\gamma z}+\pi
id,$

where $d$ is the density of the almost periodic set $A$.

Then the function $f^{\prime}(z)/f(z)$ is almost periodic on each line
$y=y_{0}\neq 0$.

Proof. Set

$\xi_{z}(t)=\begin{cases}-2\pi ie^{2\pi itz}&\text{if }t>0,\\\ 0&\text{if
}t\leq 0,\end{cases}\quad
z\in{\mathbb{C}}_{+},\quad\qquad\xi_{z}(t)=\begin{cases}2\pi ie^{2\pi
itz}&\text{if }t<0,\\\ 0&\text{if }t\geq 0,\end{cases}\quad
z\in{\mathbb{C}}_{-}.$

It is not hard to check that in the sense of distributions
$\hat{\xi}_{z}(\lambda)=1/(z-\lambda)$ for
$z\in{\mathbb{C}}_{+}\cup{\mathbb{C}}_{-}$.

Let $\varphi(t)$ be any even nonnegative $C^{\infty}$-function such that
$\operatorname{supp}\varphi\subset(-1,1)$ and $\int\varphi(t)dt=1$. Set
$\varphi_{\varepsilon}(t)=\varepsilon^{-1}\varphi(t/\varepsilon)$ for
$\varepsilon>0$. Fix $z=x+iy\in{\mathbb{C}}_{+}$. The functions
$\xi_{z}(t)\star\varphi_{\varepsilon}(t)$ and
$\hat{\xi}_{z}(\lambda)\hat{\varphi}_{\varepsilon}(\lambda)$ belong to
$S({\mathbb{R}})$. Therefore,

$(\hat{\mu},\xi_{z}\star\varphi_{\varepsilon}(t))=(\mu,\hat{\xi}_{z}(\lambda)\hat{\varphi}_{\varepsilon}(\lambda)).$

Then for any $T_{0}<\infty$ we have

$\frac{i}{2\pi}(\hat{\mu}(t),\xi_{z}\star\varphi_{\varepsilon}(t))=d\int_{-\varepsilon}^{0}e^{-2\pi
isz}\varphi_{\varepsilon}(s)ds+\sum_{0<|\gamma|\leq\varepsilon}b_{\gamma}e^{2\pi
i\gamma z}\int_{-\varepsilon}^{\gamma}e^{-2\pi isz}\varphi_{\varepsilon}(s)ds$
$+\sum_{\varepsilon<\gamma<T_{0}}b_{\gamma}e^{2\pi i\gamma
z}\int_{-\varepsilon}^{\varepsilon}e^{-2\pi
isz}\varphi_{\varepsilon}(s)ds+\sum_{\gamma\geq T_{0}}b_{\gamma}e^{2\pi
i\gamma z}\int_{-\varepsilon}^{\varepsilon}e^{-2\pi
isz}\varphi_{\varepsilon}(s)ds=I_{0}+I_{1}+I_{2}+I_{3}.$

Then we have

$\frac{i}{2\pi}(\hat{\mu},\xi_{z})=\sum_{0<\gamma\leq\varepsilon}b_{\gamma}e^{2\pi
i\gamma z}+\sum_{\varepsilon<\gamma<T_{0}}b_{\gamma}e^{2\pi i\gamma
z}+\sum_{\gamma\geq T_{0}}b_{\gamma}e^{2\pi i\gamma z}=S_{1}+S_{2}+S_{3}.$

Set $m(s)=\sum_{\gamma\in\Gamma:\,0<\gamma\leq s}|b_{\gamma}|$. Then

(19) $\sum_{\gamma\geq r}|b_{\gamma}|e^{-2\pi\gamma
y}=\int_{r}^{\infty}e^{-2\pi sy}m(ds)\leq\lim_{T\to\infty}m(T)e^{-2\pi
Ty}+2\pi y\int_{r}^{\infty}e^{-2\pi sy}m(s)ds.$

Property (15) implies $I_{3}$ and $S_{3}$ are less than given $\eta>0$ for
$T_{0}$ large enough. Taking into account that
$\sum_{-\varepsilon<\gamma<T_{0}}|b_{\gamma}|<\infty$, we get as
$\varepsilon\to 0$

$I_{0}\to d/2,\quad I_{1}\to 0,\quad S_{1}\to 0,\quad
I_{2}-S_{2}=\sum_{\varepsilon<\gamma<T_{0}}b_{\gamma}e^{2\pi i\gamma
z}\int_{-\varepsilon}^{\varepsilon}(e^{-2\pi
isz}-1)\varphi_{\varepsilon}(s)ds\to 0$

and

$\lim_{\varepsilon\to
0}(\mu,\hat{\xi}_{z}(\lambda)\hat{\varphi}_{\varepsilon}(\lambda))=(\hat{\mu},\xi_{z}(t))-2\pi
id/2=-2\pi i\sum_{\gamma\in\Gamma\cap{\mathbb{R}}_{+}}b_{\gamma}e^{2\pi
i\gamma z}-\pi id.$

On the other hand, we have

(20)
$(\mu,\hat{\xi}_{z}(\lambda)\hat{\varphi}_{\varepsilon}(\lambda))=\frac{\hat{\varphi}(\varepsilon
a_{0})}{z-a_{0}}+\sum_{n\in{\mathbb{N}}}\left[\frac{\hat{\varphi}(\varepsilon
a_{n})}{z-a_{n}}+\frac{\hat{\varphi}(\varepsilon a_{-n})}{z-a_{-n}}\right].$

The function $\hat{\varphi}(t)$ tends to $1$ as $t\to 0$ and
$|\hat{\varphi}(t)|\leq 1$. We have

$\left[\frac{\hat{\varphi}(\varepsilon
a_{n})}{z-a_{n}}+\frac{\hat{\varphi}(\varepsilon
a_{-n})}{z-a_{-n}}\right]=\hat{\varphi}(\varepsilon
a_{-n})\left[\frac{1}{z-a_{n}}+\frac{1}{z-a_{-n}}\right]+\frac{1}{z-a_{n}}[\hat{\varphi}(\varepsilon
a_{n})-\hat{\varphi}(\varepsilon a_{-n})].$

Since $\hat{\varphi}$ is even, we get with bounded $\theta(n)$ and $\phi(n)$

$\hat{\varphi}(\varepsilon a_{n})-\hat{\varphi}(\varepsilon
a_{-n})=\hat{\varphi}(\varepsilon
n/d+\varepsilon\phi(n))-\hat{\varphi}(-\varepsilon
n/d+\varepsilon\phi(-n))=\hat{\varphi}^{\prime}(\varepsilon
n/d+\varepsilon\theta(n))\varepsilon[\phi(n)+\phi(-n)].$

Since $\hat{\varphi}(t)$ belongs to Schwartz space, we see that
$\hat{\varphi}^{\prime}(t)=O(1/|t|)$ as $t\to\infty$. Hence for
$\varepsilon|n/d+\theta(n)|>1$

$|\varepsilon[\hat{\varphi}^{\prime}(\varepsilon
n/d+\varepsilon\theta(n))]|\leq C|n|^{-1}$

with a constant $C<\infty$. The same estimate (with another constant $C$) is
valid for $\varepsilon|n/d+\theta(n)|\leq 1$, that for all $n\in{\mathbb{N}}$
and $\varepsilon>0$

$|\hat{\varphi}(\varepsilon a_{n})-\hat{\varphi}(\varepsilon
a_{-n})|\leq(C/n)2\sup_{n}|\phi(n)|.$

Hence the right-hand side of (20) for all $\varepsilon>0$ is majorized by the
sum

(21)
$\frac{1}{|z-a_{0}|}+\sum_{n\in{\mathbb{N}}}\left|\frac{1}{z-a_{n}}+\frac{1}{z-a_{-n}}\right|+\sum_{n\in{\mathbb{N}}}\frac{C^{\prime}}{n|z-a_{n}|}.$

By Theorem 1, we have $1/(z-a_{n})=O(1/n)$. Taking into account also Corollary
1, we get the convergence of both sums in (21). Therefore we can pass to the
limit in (20) as $\varepsilon\to 0$ and obtain (17).

By (19), $\sum_{\gamma\geq 1}|b_{\gamma}|e^{-2\pi\gamma y_{0}}<\infty$ for
$y_{0}>0$, and $\sum_{0<\gamma<1}|b_{\gamma}|<\infty$. Therefore the series in
right-hand part of (17) absolutely and uniformly for $\mbox{Im }z\geq\alpha>0$
converges, and $f^{\prime}(z)/f(z)$ is almost periodic on the line $y=y_{0}$.

In the case $y_{0}<0$ we apply (17) to the function $\overline{f(\bar{z})}$
and obtain (18).

######  Theorem 3.

If

(22) $\int_{0}^{1}s^{-2}|\hat{\mu}|(0,s)ds<\infty,$

then the function $f$ from (16) with the zero set $A$ can be rewritten in the
form

$f(z)=\sum_{\omega\in\Omega}q_{\omega}e^{2\pi i\omega z}\qquad
q_{\omega}\in{\mathbb{C}}\setminus\\{0\\},\quad\omega\in{\mathbb{R}},\quad\sum_{\omega\in\Omega}|q_{\omega}|<\infty,$

where a countable bounded set $\Omega$ satisfies the conditions
$\sup\Omega\in\Omega$, $\inf\Omega\in\Omega$.

Proof. The sum in the right-hand side of (17) converges absolutely and
uniformly in $x\in{\mathbb{R}}$ and $y\geq\alpha>0$. Changing the order of
summing and integrating, we get for $z=x+iy\in{\mathbb{C}}_{+}$

(23) $\log f(z)-\log
f(i)=\int_{i}^{z}\frac{f^{\prime}(z)}{f(z)}dz=-\sum_{\gamma\in\Gamma\cap{\mathbb{R}}_{+}}b_{\gamma}\frac{e^{2\pi
i\gamma z}-e^{-2\pi\gamma}}{\gamma}-id\pi z-\pi d.$

It is easy to check that the convergence of the integral in (22) implies (in
fact is equivalent to) the convergence of the series

$\sum_{0<\gamma<1}|b_{\gamma}|\gamma^{-1}=\int_{0}^{1}s^{-1}d|\hat{\mu}|(0,s),$

and (19) implies

$\sum_{\gamma\geq 1}|b_{\gamma}|\gamma^{-1}e^{-2\pi\gamma}<\infty.$

Therefore,

(24) $\log f(x+i)+id\pi
x=-\sum_{0<\gamma<1}(b_{\gamma}/\gamma)e^{-2\pi\gamma}e^{2\pi i\gamma
x}-\sum_{\gamma\geq 1}(b_{\gamma}/\gamma)e^{-2\pi\gamma}e^{2\pi i\gamma
x}+C_{0}$

with some constant $C_{0}\in{\mathbb{C}}$, and $\|\log f(x+i)+id\pi
x\|_{W}<\infty$. Since $\|FG\|_{W}\leq\|F\|_{W}\|G\|_{W}$, we obtain

$f(x+i)e^{id\pi x}=\sum_{\omega\in\Omega}p_{\omega}e^{2\pi i\omega
x},\quad\sum_{\omega\in\Omega}|p_{\omega}|=\|f(x+i)e^{id\pi x}\|_{W}\leq
e^{\|\log f(x+i)+id\pi x\|_{W}}<\infty,$

with $p_{\omega}\in{\mathbb{C}}$ and a countable spectrum
$\Omega\subset{\mathbb{R}}_{+}\cup\\{0\\}$. The entire function
$f(z+i)e^{id\pi z}$ has exponential growth, therefore, by §1, Ch.VI from [10],
$\Omega$ is bounded. Hence the function

$f(z)=\sum_{\omega\in\Omega}p_{\omega}e^{\pi(2\omega-d)}e^{\pi i(2\omega-d)z}$

is also the Dirichlet series and $\|f\|_{W}<\infty$. Moreover, all zeros of
$f$ are real, therefore, by Cor.2, Ch.VI from [10], we have
$\sup\Omega\in\Omega$, $\inf\Omega\in\Omega$.

Set

$g(z):=\sum_{\gamma\in\Gamma,0<\gamma<1}b_{\gamma}\frac{e^{2\pi i\gamma
z}-1}{\gamma}.$

Since the sum $\sum_{\gamma\in\Gamma,0<\gamma<1}|b_{\gamma}|$ is bounded and

$g(z)=\sum_{k=1}^{\infty}\frac{(2\pi
iz)^{k}}{k!}\sum_{\gamma\in\Gamma,0<\gamma<1}\gamma^{k-1}b_{\gamma},$

we see that $g(z)$ is a well-defined entire function. Then for
$z\in{\mathbb{C}}_{+}\cup{\mathbb{R}}$

(25)
$|g(z)|\leq\left[\sum_{\gamma\in\Gamma,0<\gamma<\varepsilon}+\sum_{\gamma\in\Gamma,\varepsilon\leq\gamma<1}\right]\left|\frac{1-e^{2\pi
i\gamma z}}{\gamma z}\right||z||b_{\gamma}|\leq
4\pi|z|\sum_{0<\gamma<\varepsilon}|b_{\gamma}|+\frac{2}{\varepsilon}\sum_{\varepsilon\leq\gamma<1}|b_{\gamma}|.$

The sum $\sum_{0<|\gamma|<\varepsilon}|b_{\gamma}|$ is arbitrary small for
small $\varepsilon$, therefore, $|g(z)|=o(|z|)$ as
$|z|\to\infty,\,z\in{\mathbb{C}}_{+}\cup{\mathbb{R}}$. Besides, in every strip
$\\{z:\,|\mbox{Im }z|<M\\}$

(26)
$|g(z)-g(x)|\leq\sum_{\gamma\in\Gamma,0<\gamma<1}|b_{\gamma}|\left|e^{2\pi
i\gamma x}\frac{e^{-2\pi\gamma y}-1}{\gamma}\right|<C(M).$

######  Theorem 4.

If $g(z)$ is uniformly bounded for $z=x\in{\mathbb{R}}$, then the function
(16) with the zero set $A$ is almost periodic of the exponential type $\pi d$.

Conversely, if $A$ is the zero set of an entire almost periodic function of
exponential growth, then the function $g(z)$ is uniformly bounded on
${\mathbb{R}}$.

Proof. By (19), the sum $\sum_{\gamma\geq
1}b_{\gamma}\gamma^{-1}e^{-2\pi\gamma}$ is finite, as is the sum
$\sum_{0<\gamma<1}b_{\gamma}\frac{1-e^{-2\pi\gamma}}{\gamma}$. Hence we can
rewrite (23) in the form

(27) $\log f(z)=-\sum_{0<\gamma<1}b_{\gamma}\frac{e^{2\pi i\gamma
z}-1}{\gamma}-\sum_{\gamma\geq 1}b_{\gamma}\frac{e^{2\pi i\gamma
z}}{\gamma}-id\pi z+\operatorname{const}.$

Here the second sum is uniformly bounded in every half-plane $\mbox{Im
}z\geq\alpha>0$. If the function $g(x)$ is uniformly bounded on
${\mathbb{R}}$, then by (26), the first sum in (27) is bounded on every line
$\mbox{Im }z=M>0$. Therefore the functions $\log|f(z)|$ and $f(z)$ are bounded
on this line, and $f(z)$ is bounded on every line $\mbox{Im }z=-M<0$ as well.
Since the function $f(z)$ has the exponential growth, the Fragment-Lindelof
Theorem implies boundedness of $f$ in every horizontal strip of bounded width.

Furthermore, combining (27) and (25), we get

$\lim_{y\to+\infty}y^{-1}\log|f(iy)|=\pi d.$

The function $f(z)$ is bounded on the real line, therefore the Fragment-
Lindelof Theorem implies that $|f(z)|\leq Me^{y\pi d}$ for all
$z\in{\mathbb{C}}_{+}$. Also, $|f(z)|\leq Me^{-y\pi d}$ for all
$z\in{\mathbb{C}}_{-}$, so $f(z)$ has the exponential type $\pi d$.

Next, the function $\log f(z)+id\pi z$ is bounded on the line $\mbox{Im }z=1$.
Its derivative $(\log f(x+i))^{\prime}+id\pi$ is almost periodic, hence, by
Bohr’s Theorem (cf.[11], Theorem 1.2.1), the functions $\log f(x+i)+id\pi x$
and $f(x+i)$ are almost periodic in the variable $x$. By Theorem 1.2.3 from
[11], the function $f(z)$ is almost periodic in every strip, where it is
bounded, hence it is almost periodic in ${\mathbb{C}}$.

Now let $G(z)$ be an entire almost periodic function of exponential type with
zero set $A$. Clearly, $G(z)=K_{1}e^{K_{2}z}f(z)$ with
$K_{1},\,K_{2}\in{\mathbb{C}}$. Taking into account that the second sum in
(27) is bounded on the line $\mbox{Im }z=1$, we obtain as $x\to\infty$

$\log G(x+i)=K_{2}x-g(x+i)-id\pi x+O(1).$

Since $G(x+i)$ is almost periodic, left-hand side of this equality is also
uniformly bound. Then (26) implies

$K_{2}x-g(x)-id\pi x=O(1).$

By (25), $g(x)=o(|x|)$, therefore, $K_{2}=i\pi d$ and $g(x)$ is bounded.

I would like to thank the Department of Mathematics and Computer Science of
the Jagiellonian University for their hospitality and Professor Lukasz
Kosinski for his interest in my work.

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