# Non-negative crystalline and Poisson measures in the Euclidean space

Sergii Yu. Favorov  Sergii Favorov,
iii V.N.Karazin Kharkiv National University
iii Svobody sq., 4, Kharkiv, Ukraine 61022<EMAIL_ADDRESS>
> Abstract. We study properties of temperate non-negative purely atomic
> measures in the Euclidean space such that the distributional Fourier
> transform of these measures are pure point ones. A connection between these
> measures and almost periodicity is shown, several forms of the uniqueness
> theorem are proved. We also obtain necessary and sufficient conditions for a
> measure with positive integer masses on the real line to correspond the zero
> set of an absolutely convergent Dirichlet series with bounded spectrum.
>
> AMS Mathematics Subject Classification: 30B50, 42A38, 52C23
>
> Keywords: Poisson measure, crystalline measure, lighthouse, Fourier
> transform, almost periodic measure, almost periodic set, zero set, Dirichlet
> series

##  1\. Introduction

Following [20], we say that a pure point complex measure $\mu$ on
${\mathbb{R}}^{d}$ is a Poisson measure if $\mu$ is a temperate distribution
and its Fourier transform $\hat{\mu}$ is also a pure point measure. If $\mu$
and $\hat{\mu}$ have locally finite supports, then $\mu$ is said to be a
crystalline measure. If, in addition, the measures $|\mu|$ and $|\hat{\mu}|$
are temperate distributions, then $\mu$ is called the Fourier quasicrystal.

Fourier quasicrystals and crystalline measures are currently being studied
very actively. Many works are devoted to the study of the properties of
Fourier quasicrystals. For example, see the collections of papers [2], [24],
in particular, the basic work [17]. Fourier quasicrystals have applications in
modern physics, where they are used as mathematical models of certain atomic
structures.

Measures of the form

(1) $\mu_{A}=\sum_{n}\delta_{a_{n}},\quad
A=\\{a_{n}\\}\subset{\mathbb{R}}^{d},$

are the most important case of Fourier quasicrystals. A nontrivial example of
such measures in the case $d=1$, where the sequence $A$ is not a finite union
of shifts of an arithmetical progression, was found by P.Kurasov and P.Sarnak
[15]. Then A.Olevskii and A.Ulanovskii in [22], [23] showed that a sequence
$A\subset{\mathbb{R}}$, which generate Fourier quasicrystals (1) is the zero
set of some exponential polynomial

(2) $P(z)=\sum_{1\leq j\leq N}q_{n}e^{2\pi iz\omega_{j}},\qquad
q_{j}\in{\mathbb{C}},\quad\omega_{j}\in{\mathbb{R}},$

with real zeros and, conversely, every zero set $A$ of each exponential
polynomial (2) with only real zeros generates Fourier quasicrystal (1). A
multidimensional version of the last result was obtained by W.Lawton and
A.Tsikh [18].

Here we suppose that $A$ is a multiset, that is, the multiplicity of each
point $a_{n}$ in (1) (the mass of $\mu_{A}$ at $a_{n}$) is finite, and equals
the multiplicity of zero $P(z)$ at this point.

However, the condition that the support of $\hat{\mu}$ is locally finite often
too restrictive, so more general objects, Poisson measures, arise (cf. [21]).
Note that the class of Poisson measures is the widest class inducing the
general Poisson formula: if

$\mu=\sum_{\lambda\in\Lambda}a_{\lambda}\delta_{\lambda}$

and

$\hat{\mu}=\sum_{\gamma\in\Gamma}b_{\gamma}\delta_{\gamma}$

are measures on ${\mathbb{R}}^{d}$ with countable
$\Lambda,\,\Gamma\subset{\mathbb{R}}^{d}$, we have for every Schwartz function
$\varphi$

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

Most effects with Fourier quasicrystals are associated with their almost
periodicity, so our study of Poisson measures will begin with the following
theorem:

######  Theorem 1.

For every non-negative Poisson measure $\mu$ on ${\mathbb{R}}^{d}$ both
convolutions $\mu\star\psi$ with any continuous compactly supported function
$\psi$ and $\hat{\mu}\star\varphi$ with any Schwartz function $\varphi$ are
almost periodic functions in the sense of Bohr.

Using definitions of the next section, the measure $\mu$ is almost periodic,
and the measure $\hat{\mu}$ is an almost periodic distribution.

Note that every complex Fourier quasicrystal is an almost periodic
distribution [8, Lemma 1]. But there are crystalline measures that are not
almost periodic distributions [6].

Theorem 1 implies various properties of non-negative Poisson measures.

Let

(3) $B(x_{n},R_{n})=\\{x\in{\mathbb{R}}^{n}:\,|x-x_{n}|<R_{n}\\},\quad
n=1,2,\dots$

be a sequence of arbitrary balls such that $R_{n}\to\infty$ as $n\to\infty$.

######  Theorem 2.

Let $\mu,\,\nu$ be two non-negative Poisson measures on ${\mathbb{R}}^{d}$. If
either $|\mu-\nu|B(x_{n},R_{n})=o(R^{d}_{n})$, or
$|\hat{\mu}-\hat{\nu}|B(x_{n},R_{n})=o(R_{n}^{d})$ as $n\to\infty$, then
$\mu\equiv\nu$.

######  Corollary 1.

Let $\mu$ be a non-negative Poisson measure on ${\mathbb{R}}^{d}$. If
$\mu(B(x_{n},R_{n}))=o(R^{d}_{n})$ or
$|\hat{\mu}|(B(x_{n},R_{n})=o(R_{n}^{d})$ as $n\to\infty$, then $\mu\equiv 0$.

In particular, condition of this theorem are fulfilled when
$\operatorname{supp}\hat{\mu}$ does not intersect with an open cone in
${\mathbb{R}}^{d}$, hence any lighthouse $(\mu,S)$ (see [21]) with a closed
cone $S$ could not be a Poisson measure.

Let $E=\cup_{n}B(x_{n},R_{n})$ be the union of balls (3), where
$R_{n}\to\infty$ as $n\to\infty$.

######  Theorem 3.

Let

$\mu_{A}=\sum_{a_{n}\in A}\delta_{a_{n}},\quad\mu_{C}=\sum_{c_{n}\in
C}\delta_{c_{n}},\quad A,\ C\subset{\mathbb{R}}^{d},$

be two Poisson measures such that under appropriate numbering $a_{n}-c_{n}\to
0$ as $n\to\infty,\,n\in H$, where $H=\\{n\in{\mathbb{N}}:\,a_{n}\in E\
\text{or}\ c_{n}\in E\\}$. Then $\mu_{A}\equiv\mu_{C}$.

Other forms of uniqueness theorems can be found in [5]. They were formulated
for complex Fourier quasicrystals, but they are actually valid for Poisson
measures $\mu$, for which $|\hat{\mu}|$ are temperate distributions. Indeed,
the last condition implies that $\mu$ is an almost periodic distributions (see
[8, Lemma 1]), and all proofs are based only on this property.

######  Theorem 4.

Let $\mu$ be a non-negative Poisson measure on ${\mathbb{R}}^{d}$, let $E$ be
as above, and assume that for all $\lambda,\,\lambda^{\prime}\in
E\cap\operatorname{supp}\mu,\,\lambda\neq\lambda^{\prime}$ the conditions
$\mu(\lambda)\geq c>0$ and $|\lambda-\lambda^{\prime}|\geq\beta>0$ hold.
Assume also that

(4) $|\hat{\mu}|(B(0,r))=O(r^{d})\qquad(r\to\infty).$

Then for some $\lambda_{j}\in\operatorname{supp}\mu$ and nondegenerated linear
operators $T_{j}$ in ${\mathbb{R}}^{d}$

(5)
$\operatorname{supp}\mu=\cup_{j=1}^{N}T_{j}{\mathbb{Z}}^{d}+\lambda_{j},\qquad\mu=\sum_{j=1}^{N}\sum_{\lambda\in
T_{j}{\mathbb{Z}}^{d}+\lambda_{j}}Q_{j}(\lambda)\delta_{\lambda},$

where $Q_{j}$ are Dirichlet series of the form

$Q_{j}(x)=\sum_{\omega\in\Omega_{j}}q_{\omega}^{(j)}e^{2\pi
ix\cdot\omega},\qquad
q_{\omega}^{(j)}\in{\mathbb{C}},\quad\sum_{\omega\in\Omega_{j}}|q_{\omega}^{(j)}|<\infty,\quad\Omega_{j}\
\text{are bounded subsets of }\ {\mathbb{R}}^{d}.$

If we replace the condition $\mu(\lambda)\geq c>0$ with $|\mu(\lambda)|\geq
c>0$ for $\lambda\in E\cap\operatorname{supp}\mu$, then (5) is also true for
complex Poisson measures.

Here we set

$x\cdot\omega=x_{1}\omega_{1}+\dots x_{d}\omega_{d}\quad\text{for}\quad
x=(x_{1},\dots,x_{d})\in{\mathbb{R}}^{d},\
\omega=(\omega_{1},\dots,\omega_{d})\in{\mathbb{R}}^{d}.$

It was shown in [9] that for $d=1$ the zero set $A=\\{a_{n}\\}$ of Dirichlet
series (2) under condition $A\subset{\mathbb{R}}$ generates a Poisson measure
$\mu_{A}=\sum_{n}\delta_{a_{n}}$. Here we prove the precise result for
dimension $1$:

######  Theorem 5.

In order that a sequence $A=\\{a_{n}\\}\subset{\mathbb{R}}$, which generate
the measure $\mu_{A}$ in (1) be the zero set of some Dirichlet series

(6) $Q(x)=\sum_{\omega\in\Omega}q_{\omega}e^{2\pi ix\omega},\qquad
q_{\omega}\in{\mathbb{C}},\quad\sum_{\omega\in\Omega_{j}}|q_{\omega}|<\infty,\quad\Omega\
\text{is bounded countable subsets of }\ {\mathbb{R}},$

with real zeros it is necessary and sufficient that the Fourier transform
$\hat{\mu}_{A}$ be a pure point measure

(7) $\hat{\mu}_{A}=\sum_{\gamma\in\Gamma}b_{\gamma}\delta_{\gamma}$

such that

(8)
$\int_{0}^{1}t^{-1}|\hat{\mu}_{A}|(dt)=\sum_{0<\gamma<1}\frac{|b_{\gamma}|}{\gamma}<\infty$

and

(9) $\log|\hat{\mu}|(B(0,r))=o(r)\qquad(r\to\infty).$

Here a multiplicity of every zero $a_{n}$ of $Q(x)$ coincides with the mass of
the measure $\mu_{A}$ at this point.

For the part of sufficiently condition (9) can be replaced with
$\log|\hat{\mu}|(B(0,r))=O(r)$ .

Note that a non-negative measure is a temperate distribution if and only if
its masses in the balls $B(0,r)$ grow at most polynomially as $r\to\infty$
(cf.[5]). Hence condition (9) is weaker than $|\hat{\mu}|$ to be a temperate
distribution.

The article is structured as follows.

In Section 2 we give all necessary definitions and formulate auxiliary
theorems on almost periodic functions and sets, Fourier transforms, Dirichlet
series, entire functions of exponential growth.

In Section 3 we show some properties of temperate measures and prove Theorems
1–4.

In Section 4 we prove the sufficiency part of Theorem 5.

In Section 5 we prove the necessity part of Theorem 5.

##  2\. Preliminary results

The Fourier transform in the sense of distributions

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

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

where

$|x|=(x_{1}^{2}+\dots+x_{d}^{2})^{1/2},\
k=(k_{1},\dots,k_{d})\in({\mathbb{N}}\cup\\{0\\})^{d},\
\|k\|=k_{1}+\dots+k_{d},\
D^{k}=\partial^{k_{1}}_{x_{1}}\dots\partial^{k_{d}}_{x_{d}}.$

These norms generate the topology on $S({\mathbb{R}}^{d})$. Elements of the
space $S^{*}({\mathbb{R}}^{d})$ of continuous linear functionals on
$S({\mathbb{R}}^{d})$ are called temperate distributions. If $\mu\in
S^{*}({\mathbb{R}}^{d})$ is a measure, we will say that $\mu$ is a temperate
measure. The Fourier transform of a temperate distribution $f$ is defined by
the equality

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

where

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

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

Dirichlet series on ${\mathbb{R}}^{d}$

Denote by $\mathcal{H}$ the algebra of all Dirichlet series

$Q(x)=\sum_{n}r_{n}e^{2\pi i\omega_{n}\cdot
x},\,\omega_{n}\in{\mathbb{R}}^{d},$

with finite Wiener’s norm $\|Q\|_{W}=\sum_{n}|r_{n}|$.

For any $Q\in\mathcal{H}$ and any analytic function $h(z)$ on a neighborhood
of the set $\overline{\\{Q(x):\,x\in{\mathbb{R}}^{d}\\}}$ we have
$h(Q(x))\in\mathcal{H}$ (see [26], Ch.VI). In particular,

for every $Q\in\mathcal{H}$ we have $\exp Q\in\mathcal{H}$,

if $\inf_{{\mathbb{R}}^{d}}|Q(x)|>0$, we have $1/Q\in\mathcal{H}$,

if $|Q(x)|\leq C<1$, we have $\log(1+Q(x))\in\mathcal{H}$.

Entire functions of exponential growth

Zeros $a_{n}$ of any entire function $g(z),\,z\in{\mathbb{C}}$, of exponential
growth satisfy the conditions

(10) $\\#\\{a_{n}:\,|a_{n}|\leq r\\}=O(r)\quad\text{as}\quad r\to\infty,$

(as usual, $\\#E$ means the number of points of a finite multiset $E$, taking
into account a multiplicity of each point), and

(11) $\sum_{n:\,|a_{n}|\leq r,a_{n}\neq
0}\frac{1}{a_{n}}=O(1)\quad\text{as}\quad r\to\infty.$

On the other hand, if a sequence
$A=\\{a_{n}\\}\subset{\mathbb{C}}\setminus\\{0\\}$ satisfies these conditions,
then the function

(12) $F(z)=\prod_{n=-\infty}^{\infty}(1-z/a_{n})e^{z/a_{n}}$

is an entire function of exponential growth (Lindelöf’s Theorem, see [14]).
Note that each $a\in{\mathbb{C}}$ can be repeated in the sequence
$\\{a_{n}\\}$ any finite number of times.

If a function $g(z)$ of the exponential type $\sigma$ is bounded on the real
axis, then for all $z\in{\mathbb{C}}$

$|g(z)|\leq e^{\sigma|\mbox{Im }z|}\sup_{z\in{\mathbb{R}}}|g(z)|\quad\text{
(Phragment--Lindel\"{o}f Principle, see \cite[cite]{[\@@bibref{}{K}{}{}]})}.$

Almost periodic functions and measures

######  Definition 1 (cf. [1], [19]).

A continuous function $g(x)$ on ${\mathbb{R}}^{d}$ is almost periodic if for
any $\varepsilon>0$ the set of $\varepsilon$-almost periods

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

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

For example every sum

(13) $Q(x)=\sum q_{n}e^{2\pi
ix\cdot\omega_{n}},\quad\omega_{n}\in{\mathbb{R}}^{d},\quad
q_{n}\in{\mathbb{C}},$

with finite Wiener’s norm $\|Q\|_{W}=\sum_{n}|q_{n}|$ is an almost periodic
function. Note that each almost periodic function is uniformly bounded and
uniformly continuous. Also, a uniform in ${\mathbb{R}}^{d}$ limit of a
sequence of almost periodic functions is almost periodic.

Set $U(T)=\\{x=(x_{1},\dots,x_{d})\in{\mathbb{R}}^{d}:\,-T/2<x_{j}\leq
T/2,\,j=1,\dots,d\\}$. We can prove the existence of the mean value of any
almost periodic function

$Mg=\lim_{T\to\infty}\frac{1}{T^{d}}\int_{U(T)}g(x)dx$

the same method as in the dimension $1$ in [1], also see [10]. The Fourier
coefficient

$b_{\omega}(g)=\lim_{T\to\infty}\frac{1}{T^{d}}\int_{U(T)}g(x)e^{-2\pi
i\omega\cdot x}dx.$

is assigned to every almost periodic function $g$ and every
$\omega\in{\mathbb{R}}^{d}$; spectrum of $g$ is the set

$\operatorname{sp}g=\\{\omega:\,b_{\omega}(g)\neq 0\\}.$

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

######  Definition 2 (cf.[25]).

A measure $\mu$ is almost periodic, if for any continuous function $\varphi$
with compact support the convolution of
$\mu\star\varphi=\int\varphi(t-x)\mu(dx)$ is almost periodic in
$t\in{\mathbb{R}}^{d}$.

######  Definition 3 (cf.[25]).

A distribution $F$ is almost periodic, if for any function $\varphi\in
S({\mathbb{R}}^{d})$ the convolution of $\varphi\star F=(F(x),\varphi(t-x))$
is almost periodic in $t\in{\mathbb{R}}^{d}$.

Note that there exist non-almost periodic measures that are almost periodic in
sense of distributions ([21], [11]).

Almost periodic sets

Here we give the definition of almost periodic sets only for sets on
${\mathbb{R}}$. The original geometric definition for sets on any horizontal
strip in ${\mathbb{C}}$ see [16, Appendix VI], [27]. Comparison of various
definitions of almost periodic sets see [12]. Almost periodic sets in
${\mathbb{R}}^{d}$ were considered in [10], [11].

######  Definition 4 ([12]).

Let $A=\\{a_{n}\\}\subset{\mathbb{R}}$ be a sequence without finite limit
points. If the convolution of $\mu_{A}=\sum_{n}\delta_{a_{n}}$ with any
$C^{\infty}$-function with compact support is almost periodic, then $A$ is
called an almost periodic set.

It is easy to see that any almost periodic set is relatively dense. Then every
point $x\in{\mathbb{R}}$ can appears in $A$ any finite number of times, hence
almost periodic sets are in fact multisets.

A zero set of every Dirichlet series (13) with only real zeros is almost
periodic [16]. 111In fact, the zero set of any almost periodic holomorphic
function in a strip in ${\mathbb{C}}$ is almost periodic. Moreover, the
divisor of any almost periodic holomorphic function in a tube subdomain in
$C^{d}$ is also almost periodic. The corresponding definitions of almost
periodic functions and sets in a strip and connections between these objects
see [1], [16], [27], [3]. Multidimensional case see [25], [13], [4].

Let $A\subset{\mathbb{R}}$ be an almost periodic set. Take any
$C^{\infty}$-function $\varphi\geq 0$ with compact support such that
$\varphi(x)\equiv 1$ for $0<x<1$. Since $\mu_{A}\star\varphi$ is almost
periodic, it is uniformly bounded. We get

######  Proposition 1.

Numbers $\\#(A\cap(x,x+1))$ are uniformly bounded in $x\in{\mathbb{R}}$.

Since any continuous function $g$ with compact support can be approximated in
the uniformly metric with a sequence $\varphi_{n}\in C^{\infty}$ with compact
supports such that
$\cap_{n}\operatorname{supp}\varphi_{n}=\operatorname{supp}g$, it follows from
Proposition 1 that we can replace $C^{\infty}$-functions by continuous
functions with compact support in Definition 4.

Just as the existence of the mean value of an almost periodic function in [1]
was proven, using Proposition 1 it is easy to show the existence of the
density

$D=\lim_{l\to\infty}\frac{\\#A\cap(x,\,x+l)}{l}$

uniformly with respect to $x\in{\mathbb{R}}$ (cf.[10]). Clearly, $D>0$.

It follows from [20, Lemma 5.5] that in the case
$\hat{\mu}_{A}=\sum_{\gamma\in\Gamma}b_{\gamma}\delta_{\gamma}$ we get
$D=b_{0}$.

######  Proposition 2 ([9]).

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

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

Conversely, any set $A=\\{kn+\phi(n)\\}$, $\phi$ as above, form an almost
periodic set with density $1/k$.

##  3\. Properties of temperate measures and proofs of theorems 1–4

######  Proposition 3.

[8, Lemma 1]. Let $\mu$ be a temperate measure and $\hat{\mu}$ be measure such
that $|\hat{\mu}|\in S^{*}({\mathbb{R}}^{d})$. Then $\mu$ is an almost
periodic distribution.

The condition $|\hat{\mu}|\in S^{*}({\mathbb{R}}^{d})$ is equivalent to the
polynomially growth of masses $|\hat{\mu}|$ in the balls $B(0,r)$ as
$r\to\infty$ (cf.[5]).

We will say that a measure $\mu$ is translation bounded, if
$\sup_{x\in{\mathbb{R}}^{d}}|\mu|B(x,1)<\infty$. It is easy to see that any
translation bounded measure is a temperate measure. Every almost periodic
measure is translation bounded (see [25] or [19]).

######  Proposition 4.

[20, Lemma 5.8]. If $\mu$ is a translation bounded measure and $\hat{\mu}$ is
a pure point measure, then $\mu$ is almost periodic.

This Meyer’s proposition is of fundamental importance in our investigation.
For convenience of readers, we give its complete proof here.

Proof. Let $\hat{\mu}=\sum_{\gamma\in\Gamma}b_{\gamma}\delta_{\gamma}$ and
$\varphi$ be any $C^{\infty}$-function on ${\mathbb{R}}^{d}$ with compact
support. We have

$\mu\star\check{\varphi}(t)=(\hat{\mu}(x),\varphi(x)e^{2\pi it\cdot
x})=\sum_{\gamma\in\Gamma\cap\operatorname{supp}\varphi}b_{\gamma}\varphi(\gamma)e^{2\pi
it\cdot\gamma}$

Since $\hat{\mu}$ is a measure, we get

$\sum_{\gamma\in\Gamma\cap\operatorname{supp}\varphi}|b_{\gamma}||\varphi(\gamma)|\leq\sup|\varphi(\gamma)|\sum_{\gamma\in\Gamma\cap\operatorname{supp}\varphi}|b_{\gamma}|<\infty.$

Hence the function $\mu\star\check{\varphi}(t)$ is almost periodic.

Now let $\phi$ is an arbitrary function from $S({\mathbb{R}}^{d})$. Since
$\hat{\phi}\in S({\mathbb{R}}^{d})$, we can approximate the function
$\check{\phi}(x)=\hat{\phi}(-x)$ with compactly supported
$C^{\infty}$-functions $\varphi_{n}$ in the metric of the space
$S({\mathbb{R}}^{d})$. Therefore the functions $\hat{\varphi}_{n}$ approximate
the function $\phi$ in the same metric. Also,

$|\phi(x)-\hat{\varphi}_{n}(x)|\leq
N_{d+1}(\phi-\hat{\varphi}_{n})\min\\{1,|x|^{-d-1}\\}.$

Therefore,

$|\mu\star\phi(t)-\mu\star\hat{\varphi}_{n}(t)|\leq
N_{d+1}(\phi-\hat{\varphi}_{n})\int\min\\{1,|x-t|^{-d-1}\\}|\mu|(dx)=$
$N_{d+1}(\phi-\hat{\varphi}_{n})|\mu|B(t,1)+N_{d+1}(\phi-\hat{\varphi}_{n})\int_{1}^{\infty}r^{-d-1}d|\mu|(B(t,r)).$

Since the measure $\mu$ is translation bounded, we see that $|\mu|B(t,1)\leq
C_{0}$ and $|\mu|B(t,r)\leq C_{d}r^{d}$ as $r>1$ uniformly with respect to
$t\in{\mathbb{R}}^{d}$. Therefore the both terms tend to $0$ uniformly with
respect to $t$ as $\hat{\varphi}_{n}\to\phi$ in the space
$S({\mathbb{R}}^{d})$, and the convolution $\phi\star\mu(t)$ is almost
periodic. For any continuous function $\psi$ with compact support there are
functions $\phi_{n}\in S({\mathbb{R}}^{d})$ with compact supports, which
uniformly converge to $\psi$ and
$\cap_{n}\operatorname{supp}\phi_{n}=\operatorname{supp}\psi$. Taking into
account that $\mu$ is translation bounded, we get again that the functions
$\phi_{n}\star\mu(t)$ tend to $\psi\star\mu(t)$ uniformly in
$t\in{\mathbb{R}}^{d}$. Hence the last function is almost periodic too, and
$\mu$ is an almost periodic measure.

Proof of Theorem 1. Let $\phi$ is a $C^{\infty}$-function with compact support
in ${\mathbb{R}}^{d}$ and non-negative Fourier transform $\Psi(x)$ (to
construct such $\phi$ should be taken
$\phi(t)=\int\bar{\varphi}(s-t)\varphi(s)ds$ for $C^{\infty}$-function
$\varphi$ with compact support). There exists a ball $B(a,\rho)$ such that
$\inf_{x\in B(a,\rho)}\Psi(x)\geq\eta>0$. Therefore,

$\eta\mu(B(a,\rho))\leq\int_{B(a,\rho)}\Psi(t)\mu(dt)\leq\int_{{\mathbb{R}}^{d}}\Psi(t)\mu(dt)=\left|\int_{{\mathbb{R}}^{d}}\phi(y)\hat{\mu}(dy)\right|\leq\int_{\operatorname{supp}\phi}|\phi(y)||\hat{\mu}|(dy).$

Then for any $x\in{\mathbb{R}}^{d}$

$\eta\mu(B(x,\rho))\leq\int_{{\mathbb{R}}^{d}}\Psi(t+a-x)\mu(dt)=\left|\int_{{\mathbb{R}}^{d}}\phi(y)e^{2\pi
iy(a-x)}\hat{\mu}(dy)\right|\leq\int_{\operatorname{supp}\phi}|\phi(y)||\hat{\mu}|(dy).$

Hence, values $\mu(B(x,\rho)))$ are uniformly bounded, and the measure $\mu$
is translation bounded. Proposition 4 implies almost periodicity of $\mu$.
Since $\mu$ is translation bounded, we see that $\mu$ is a temperate
distribution, hence by Proposition 3, the convolution $\hat{\mu}\star\varphi$
is almost periodic for every $\varphi\in S({\mathbb{R}}^{d})$.

Proof of Theorem 2. Assume that $\mu\not\equiv\nu$. Then there is a point
$x\in{\mathbb{R}}^{d}$ and a continuous function $\varphi,\,0\leq\varphi\leq
1$, with $\operatorname{supp}\varphi\subset B(0,1)$ such that
$\mu\star\varphi(x)\neq\nu\star\varphi(x)$. Let
$2\varepsilon<|\mu\star\varphi(x)-\nu\star\varphi(x)|$. The function
$g=\mu\star\varphi-\nu\star\varphi$ is almost periodic, hence for
$r_{\varepsilon}$ large enough every ball $B(x^{\prime},r_{\varepsilon})$
contains the ball $B(x+\tau,1)$ for some $\varepsilon$-almost period $\tau$ of
the function $g$. Therefore, $|g(x+\tau)|>\varepsilon$ and

$|\mu-\nu|B(x^{\prime},r_{\varepsilon})\geq|\mu-\nu|\star\varphi(x+\tau)\geq|\mu\star\varphi(x+\tau)-\nu\star\varphi(x+\tau)|\geq\varepsilon.$

Note that every ball of radius $R$ contains at least
$\kappa_{d}(R/r_{\varepsilon})^{d}$ disjointed balls of radius
$r_{\varepsilon}$, where the constant $\kappa_{d}$ depends only on the
dimension $d$. Therefore,

$|\mu-\nu|B(x_{n},R_{n})\geq\kappa_{d}r_{\varepsilon}^{-d}\varepsilon
R_{n}^{d}.$

We get a contradiction with conditions of the theorem.

The same arguments work in the case $\hat{\mu}\not\equiv\hat{\nu}$ after
replacing a continuous $\varphi$ with $C^{\infty}$-function $\varphi$.

Proof of Theorem 3. Assume $\mu\not\equiv\nu$. Let $x$, $\varphi$,
$\varepsilon$, $r_{\varepsilon}$ be the same as above. Take appropriate $\tau$
such that $B(x+\tau,r_{\varepsilon})\subset B(x_{n},R_{n})$ for some $n$ and

(14) $|\mu\star\varphi(x+\tau)-\nu\star\varphi(x+\tau)|\geq\varepsilon.$

Let $M_{\tau}$ be a set of pairs $\\{(a_{n},c_{n})\\}$ such that either
$a_{n}\in\operatorname{supp}\mu\cap B(x+\tau,1)$ or
$c_{n}\in\operatorname{supp}\nu\cap B(x+\tau,1)$. By Theorem 1, the measures
$\mu$ and $\nu$ are translation bounded. Therefore quantities $\\#M_{\tau}$
are bounded by a constant, which does not depend on $\tau$. We have

$|\mu\star\varphi(x+\tau)-\nu\star\varphi(x+\tau)|\leq\sum_{(a_{n},c_{n})\in
M_{\tau}}|\varphi(a_{n})-\varphi(c_{n})|\leq(\\#M_{\tau})\kappa_{d}N_{1}(\varphi)\max_{(a_{n},c_{n})\in
M_{\tau}}|a_{n}-c_{n}|.$

It follows from conditions of the theorem that this inequality contradicts to
(14) for $n$ large enough.

Proof of Theorem 4. Assume that there is a point $x\in{\mathbb{R}}^{d}$ such
that $0<\mu(x)<c$. Set $\varepsilon<(1/2)\min\\{(c-\mu(x)),\mu(x)\\}$ and
$r<b/2$ such that $\mu(B(x,2r))<\mu(x)+\varepsilon$. Let $\varphi(x),\
0\leq\varphi(x)\leq 1$ be a continuous function with support in $B(0,r)$ such
that $\varphi(0)=1$. We have

$\mu(x)\leq\mu\star\varphi(x)\leq\mu(B(x,r))<\mu(x)+\varepsilon.$

For $R_{n}$ large enough the ball $B(x_{n},R_{n})\subset E$ contains the ball
$B(x+\tau,r)$ for some $\varepsilon$-almost period $\tau$ of the function
$\mu\star\varphi$. Then $\mu\star\varphi(x+\tau)>0$, hence the ball
$B(x+\tau,r)$ contains exactly one point $x^{\prime}\in\operatorname{supp}\mu$
such that $c\leq\mu(x^{\prime})\leq\mu\star\varphi(x^{\prime})$. Therefore,

$c-\varepsilon<\mu\star\varphi(x^{\prime}-\tau)\leq\mu(B(x^{\prime}-\tau,r))\leq\mu(B(x,2r))<\mu(x)+\varepsilon,$

which contradicts our choice of $\varepsilon$. Consequently, the estimate
$\mu(x)\geq c$ holds for all $x\in\operatorname{supp}\mu$.

Assume that there are points $x,\,x+h\in\operatorname{supp}\mu$ such that
$|h|<b$. Let $\varphi,\ 0\leq\varphi(x)\leq 1$ be a continuous function with
support in $B(0,\varepsilon)$ with $\varepsilon<\min\\{(b-|h|),c\\}$ such that
$\varphi(0)=1$. Then $\mu\star\varphi(x)\geq c$ and $\mu\star\varphi(x+h)\geq
c$. For $R_{n}$ large enough the ball $B(x_{n},R_{n})\subset E$ contains the
ball $B(x+\tau,|h|+\varepsilon)$ for some $\varepsilon$-almost period $\tau$
of the function $\mu\star\varphi$. Then

$\mu\star\varphi(x+\tau)>0,\qquad\mu\star\varphi(x+h+\tau)>0,$

hence there are points $x^{\prime}\in
B(x+\tau,\varepsilon),\,x^{\prime\prime}\in B(x+h+\tau,\varepsilon)$ such that
$\mu(x^{\prime})>0,\,\mu(x^{\prime\prime})>0$. Since
$|x^{\prime}-x^{\prime\prime}|<|h|+2\varepsilon<b$, we get a contradiction.

In the case of a complex measure $\mu$ condition (4) and Proposition 3 yield
that $\mu$ is an almost periodic distribution. From here, repeating the
previous arguments with replacing $\mu$ by $|\mu|$, we obtain that
inequalities $|\mu(\lambda)|\geq c$ and
$|\lambda-\lambda^{\prime}|\geq\beta>0,\,\lambda\neq\lambda^{\prime}$ hold for
all points $\lambda,\,\lambda^{\prime}\in\operatorname{supp}\mu$. Under these
conditions the first equality in (5) was proved in [8, Theorem 5], and the
second one is contained in [7, Theorem 4].

##  4\. Proof of sufficiency in Theorem 5

Let $A=\\{a_{n}\\}\subset{\mathbb{R}}$ be a sequence such that the Fourier
transform $\hat{\mu}_{A}$ is measure (7). Our theorem does not depend on
shifts along real axis, hence we can assume $a_{n}\neq 0$ for all $n$. Since
$\mu_{A}$ is a non-negative Poisson measure, we see that $A$ is an almost
periodic set. By Proposition 2, we can renumber $a_{n}$ such that
$a_{n}=n/D+\phi(n),\,n\in{\mathbb{Z}}$ with a bounded function $\phi(n)$ and a
density $D$, and we get condition (10).

Consider the sum

(15) $\sum_{n\in{\mathbb{Z}}}\frac{1}{x+iy-
a_{n}}=\sum_{n\in{\mathbb{Z}}}\frac{1}{x+iy-n/D-\phi(n)}.$

Set $M=\sup_{\mathbb{R}}|\phi(n)|$. Since

(16)
$\sum_{|n/D|<M+|x|+1}\left|\frac{1}{x+iy-n/D-\phi(n)}\right|+\sum_{|n/D|\geq
M+|x|+1}\left|\frac{1}{x+iy-n/D-\phi(n)}+\frac{1}{x+iy+n/D-\phi(-n)|}\right|\leq
R,$

with $R$ depending on $x$ and $y$, we get that (15) is finite for every
$x\in{\mathbb{R}},\,y>0$.

Furthermore, the sums

$\sum_{n\in{\mathbb{Z}},|a_{n}|<N}\frac{1}{a_{n}}\quad\mbox{and}\quad\sum_{n\in{\mathbb{Z}},|n/D|<N}\frac{1}{a_{n}}=\frac{1}{a_{0}}+\sum_{n=1}^{N}\left[\frac{1}{n/D+\phi(n)}+\frac{1}{-n/D+\phi(-n)}\right]$

differ for a uniformly bounded (with respect to $N$) number of terms, and each
of these terms tends to $0$ as $N\to\infty$. Therefore these sums have the
same finite limit $L$ as $N\to\infty$, and we get (11). By Lindelof’s Theorem,
the function $F(z)$ from (12) has exponential growth. Moreover, we obtain that
the sum

$\sum_{n\in{\mathbb{N}}}\left|\frac{1}{a_{n}}+\frac{1}{a_{-n}}\right|$

is also converges, hence

$F(z)=(1-z/a_{0})e^{z/a_{0}}\prod_{n\in{\mathbb{N}}}(1-z/a_{n})(1-z/a_{-n})e^{z(1/a_{n}-1/a_{-n})}=e^{Lz}f(z),$

where

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

is a well-defined entire function of exponential growth.

Set for $\mbox{Im }z>0$

$\xi_{z}(t)=\begin{cases}-2\pi ie^{2\pi itz}&\text{if }t>0,\\\ 0&\text{if
}t\leq 0,\end{cases}$

It is not hard to check that in the sense of distributions
$\hat{\xi}_{z}(\lambda)=1/(z-\lambda)$. Let $\varphi(t)$ be any even
nonnegative $C^{\infty}$-function such that

$\operatorname{supp}\varphi\subset(-1,1)\quad\text{and}\quad\int\varphi(t)dt=1.$

Set $\varphi_{\varepsilon}(t)=\varepsilon^{-1}\varphi(t/\varepsilon)$ for
$\varepsilon>0$. The functions $\xi_{z}(t)\star\varphi_{\varepsilon}(t)$ and
$\hat{\xi}_{z}(\lambda)\hat{\varphi}_{\varepsilon}(\lambda)$ belong to
$S({\mathbb{R}})$. Therefore,

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

We have

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

Clearly, $\hat{\varphi}_{\varepsilon}(t)\to 1$ as $\varepsilon\to 0$. Taking
into account (16), we can apply Lebesgue’s Dominate Convergence Theorem.
Hence,

(18)
$(\mu_{A},\hat{\xi}_{z}(\lambda)\hat{\varphi}_{\varepsilon}(\lambda))\to\frac{1}{z-a_{0}}+\sum_{n\in{\mathbb{N}}}\left[\frac{1}{z-a_{n}}+\frac{1}{z-a_{-n}}\right]\qquad(\varepsilon\to
0).$

Set
$n(s)=\sum_{\gamma\in\Gamma:\,0<\gamma<s}|b_{\gamma}|=|\hat{\mu}_{A}|(0,s)$.
If $|\hat{\mu}_{A}|(0,s)\leq Ce^{2\pi sK}$, we obtain for $y>K$

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

Therefore the series $\sum_{\gamma>0}b_{\gamma}e^{2\pi i\gamma z}$ absolutely
and uniformly converges for every fixed $y=\mbox{Im }z>K$.

Further, we have for $\varepsilon>0$

$\frac{i}{2\pi}(\hat{\mu}_{A}(t),\xi_{z}\star\varphi_{\varepsilon}(t))=\sum_{\gamma<0}b_{\gamma}e^{2\pi
i\gamma z}\int_{s<\gamma}e^{-2\pi
isz}\varphi_{\varepsilon}(s)ds+b_{0}\int_{s<0}e^{-2\pi
isz}\varphi_{\varepsilon}(s)ds\\\ +\sum_{\gamma>0}b_{\gamma}e^{2\pi i\gamma
z}\int_{s<\gamma}e^{-2\pi isz}\varphi_{\varepsilon}(s)ds.$

It is easy to see that all integrals are bounded and

$\int_{s<\gamma<0}e^{-2\pi isz}\varphi_{\varepsilon}(s)ds\to
0,\quad\int_{s<0}e^{-2\pi
isz}\varphi_{\varepsilon}(s)ds\to\frac{1}{2},\quad\int_{0<s<\gamma}e^{-2\pi
isz}\varphi_{\varepsilon}(s)ds\to 1\qquad(\varepsilon\to 0).$

Using Lebesgue’s Dominate Convergence Theorem, we obtain

$(\hat{\mu}_{A},\xi_{z}\star\varphi_{\varepsilon}(t))\to-2\pi
i\sum_{\gamma\in\Gamma\cap(0,+\infty)}b_{\gamma}e^{2\pi i\gamma z}-\pi
ib_{0}\qquad(\varepsilon\to 0),$

and by (18),

(20)
$f^{\prime}(z)/f(z)=\frac{1}{z-a_{0}}+\sum_{n\in{\mathbb{N}}}\left[\frac{1}{z-a_{n}}+\frac{1}{z-a_{-n}}\right]=-2\pi
i\sum_{\gamma\in\Gamma\cap(0,+\infty)}b_{\gamma}e^{2\pi i\gamma z}-\pi iD,$

where we replace $b_{0}$ with the density $D$. Therefore, $f^{\prime}(z)/f(z)$
is an absolutely convergent Dirichlet series on the line
$x\in{\mathbb{R}},\,y=\operatorname{const}>K$, and its mean value equals $-\pi
iD$.

Changing the order of summation and integration in (20), we get

(21) $\log f(x+iy)-\log
f(iy)=\int_{iy}^{x+iy}\frac{f^{\prime}(\zeta)}{f(\zeta)}d\zeta=-\sum_{\gamma\in\Gamma\cap(0,+\infty)}b_{\gamma}\frac{(e^{2\pi
i\gamma x}-1)e^{-2\pi\gamma y}}{\gamma}-iD\pi x.$

Since $|e^{2\pi i\gamma x}-1|/|\gamma|\leq 2\pi|x|$, we see that (21) is well
defined for all $x\in{\mathbb{R}}$ and $y>K$. Then

$\|\log f(x+iy)+iD\pi
x\|_{W}\leq\sum_{0<\gamma<1}\frac{|b_{\gamma}|}{\gamma}2e^{-2\pi\gamma
y}+\sum_{\gamma\geq 1}\frac{|b_{\gamma}|}{\gamma}2e^{-2\pi\gamma
y}+|\log|f(iy)||<\infty.$

Condition (8) implies that the first sum in the right-hand side converges, and
(19) implies that the second one converges as well.

Redefine $y=y_{0}$. We obtain

$f(x+iy_{0})e^{iD\pi x}=\sum_{\omega\in\Omega}\beta_{\omega}e^{2\pi i\omega
x},\quad\sum_{\omega\in\Omega}|\beta_{\omega}|=\|f(x+iy_{0})e^{iD\pi
x}\|_{W}\leq e^{\|\log f(x+iy_{0})+iD\pi x\|_{W}}<\infty,$

with $\beta_{\omega}\in{\mathbb{C}}$ and a countable spectrum $\Omega$. The
entire function $f(z+iy_{0})e^{iD\pi z}$ has the exponential growth with zeros
in a horizontal strip of bounded width. It follows from [16, §1, Ch.VI], that
$\Omega$ is bounded. Hence the function

$f(z)=\sum_{\omega\in\Omega}\beta_{\omega}e^{\pi(2\omega-D)y_{0}}e^{\pi
i(2\omega-D)z}$

is also a Dirichlet series and $\|f\|_{W}<\infty$.

##  5\. Proof of necessity in Theorem 5

Let $Q$ be Dirichlet series of the form (6) with the zero set
$A\subset{\mathbb{R}}$. Since zeros of $Q(z)$ lie in a strip of finite width,
we get that $\kappa:=\inf\Omega\in\Omega$ and
$\kappa^{\prime}:=\sup\Omega\in\Omega$ (see [16], Ch.VI, Cor.2). We obtain for
$z=x+iy$

$\log Q(x+iy)=\log q_{\kappa}+2\pi i\kappa(x+iy)+\log(1+H(x+iy)),\quad
H(x+iy)=\sum_{\omega\in\Omega\setminus\\{\kappa\\}}\frac{q_{\omega}}{q_{\kappa}e^{2\pi(\omega-\kappa)y}}e^{2\pi
i(\omega-\kappa)x}.$

Taking into account that $\sum_{\omega}|q_{\omega}|<\infty$, we can take a
finite number of points $\omega_{1},\dots,\omega_{N}\in\Omega$ and then $s>0$
such that

$\sum_{\omega\in\Omega\setminus\\{\kappa,\omega_{1},\dots,\omega_{N}\\}}|q_{\omega}/q_{\kappa}|<1/3,\qquad\sum_{j=1}^{N}e^{-2\pi(\omega_{j}-\kappa)s}|q_{\omega_{j}}/q_{\kappa}|<1/3.$

Also, we may suppose that for the same $\omega_{1},\dots,\omega_{N}$ and $s$

(22)
$\sum_{\omega\in\Omega\setminus\\{\kappa^{\prime},\omega_{1},\dots,\omega_{N}\\}}|q_{\omega}/q_{\kappa^{\prime}}|<1/3,\qquad\sum_{j=1}^{N}e^{2\pi(\omega_{j}-\kappa^{\prime})s}|q_{\omega_{j}}/q_{\kappa^{\prime}}|<1/3.$

So $\|H(x+iy)\|_{W}<2/3$ for $y\geq s$, and the function

$\log(1+H(x+iy))=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}H^{n}(x+iy)}{n}$

is well-defined and belongs to $\mathcal{H}$. Since
$\operatorname{sp}H(x+iy)\subset(0,+\infty)$, we see that spectrum of the
function

(23) $\log Q(x+iy)-2\pi i\kappa(x+iy)-\log q_{\kappa}=\log(1+H(x+iy))$

lies in $(0,+\infty)$ too.

From almost periodicity of $Q$ it follows that for any $\varepsilon>0,\,s>0$
there exists $m(\varepsilon,s)>0$ such that

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

([16], Ch.6, Lemma 1). Hence we have for any fixed $y>0$

(24) $\inf_{x\in{\mathbb{R}}}|Q(x\pm iy)|>0.$

Further, the zero set $A$ of the Dirichlet series $Q(z)$ is almost periodic,
and Proposition 1 implies that each connected component of $A(\varepsilon)$
contains no segment of length $1$ for sufficiently small $\varepsilon$. Hence
there are sequences $L_{k}\to+\infty,\,L^{\prime}_{k}\to-\infty$ and $m>0$
such that

$|Q(x+iy)|>m\quad\text{for}\quad x=L_{k}\quad\text{or}\quad
x=L^{\prime}_{k},\quad|y|\leq s.$

Let $\varphi$ be $C^{\infty}$-function with compact support on some interval
$(-r,r)$. Set

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

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

(25) $\check{\Phi}(t+iy)=\varphi(t)e^{2\pi ty},\qquad t\in{\mathbb{R}}.$

Hence the function $\Phi(x+iy)$ tends to zero for $x=L_{k}$ or
$x=L^{\prime}_{k}$ as $k\to\infty$ uniformly with respect to $y\in[-s,s]$.
Consider integrals of the function $\Phi(z)Q^{\prime}(z)Q^{-1}(z)$ over the
boundaries of rectangles
$\Pi_{k}=\\{z=x+iy:\,L^{\prime}_{k}<x<L_{k},|y|<s\\}$. These integrals tend to
the difference

(26)
$\int_{-is-\infty}^{-is+\infty}\Phi(z)Q^{\prime}(z)Q^{-1}(z)dz-\int_{is-\infty}^{is+\infty}\Phi(z)Q^{\prime}(z)Q^{-1}(z)dz=:I_{1}-I_{2}$

as $k\to\infty$. On the other hand, they tend to the sum

(27) $2\pi
i\sum_{\lambda:Q(\lambda)=0}\operatorname{Res}_{\lambda}\Phi(z)Q^{\prime}(z)Q^{-1}(z)=2\pi
i\sum_{\lambda:Q(\lambda)=0}a(\lambda)\hat{\varphi}(\lambda)=2\pi
i(\mu_{A},\hat{\varphi})=2\pi i(\hat{\mu}_{A},\varphi),$

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

We have $Q^{\prime}/Q=(\log Q)^{\prime}$, hence (23) implies

(28) $Q^{\prime}(x+is)/Q(x+is)=\sum_{\gamma\in\Gamma^{*}}p_{\gamma}e^{2\pi
i\gamma x},\quad\sum_{\gamma\in\Gamma^{*}}|p_{\gamma}|<\infty,\quad p_{0}=2\pi
i\kappa,\quad p_{\gamma}=0\quad\text{for}\quad\gamma<0.$

It follows from (22) that the same arguments show

$Q^{\prime}(x-is)/Q(x-is)=\sum_{\gamma\in\Gamma_{*}}\tilde{p}_{\gamma}e^{2\pi
i\gamma
x},\quad\sum_{\gamma\in\Gamma_{*}}|\tilde{p}_{\gamma}|<\infty,\quad\tilde{p}_{0}=2\pi
i\kappa^{\prime},\quad\tilde{p}_{\gamma}=0\quad\text{for}\quad\gamma>0.$

The functions $\Phi(x\pm is)$ belong to $S({\mathbb{R}})$, hence
$|x|^{2}\Phi(x\pm is)\to 0$ as $|x|\to\infty$. Changing the order of
integration and summation and taking into account (25), we obtain for the
difference of integrals (26)

(29)
$I_{1}-I_{2}=\sum_{\gamma\in\Gamma_{*}}\tilde{p}_{\gamma}\int_{-\infty}^{+\infty}\Phi(x-is)e^{2\pi
i\gamma
x}dx-\sum_{\gamma\in\Gamma^{*}}p_{\gamma}\int_{-\infty}^{+\infty}\Phi(x+is)e^{2\pi
i\gamma x}dx\\\ =\sum_{\gamma\in\Gamma_{*}}\tilde{p}_{\gamma}e^{-2\pi\gamma
s}\varphi(\gamma)-\sum_{\gamma\in\Gamma^{*}}p_{\gamma}e^{2\pi\gamma
s}\varphi(\gamma).$

Since $\operatorname{supp}\varphi\subset(-r,r)$, we obtain

(30)
$|(\hat{\mu}_{A},\varphi)|\leq(2\pi)^{-1}\left(\sum_{\gamma\in\Gamma^{*}}|p_{\gamma}|+\sum_{\gamma\in\Gamma_{*}}|\tilde{p}_{\gamma}|\right)e^{2\pi
rs}\sup_{|y|\leq r}|\varphi(y)|.$

Hence the distribution $\hat{\mu}_{A}$ has a unique expansion to a linear
functional on the space of continuous functions $g$ on $[-r,r]$ such that
$g(-r)=g(r)=0$ with the same bound. Since we can expand this functional to the
space of all continuous functions on $[-r,r]$ with the same bound too, we see
that $\hat{\mu}_{A}$ is a complex measure. Then (27) and (29) implies

(31)
$\hat{\mu}_{A}=\sum_{\gamma\in\Gamma_{*}}\frac{\tilde{p}_{\gamma}e^{-2\pi\gamma
s}}{2\pi
i}\delta_{\gamma}-\sum_{\gamma\in\Gamma^{*}}\frac{p_{\gamma}e^{2\pi\gamma
s}}{2\pi i}\delta_{\gamma}.$

Therefore, $\hat{\mu}_{A}$ is a purely point measure. Also, it follows from
(30) that

(32) $|\hat{\mu}_{A}|(-r,r)\leq C(s)e^{2\pi sr}.$

Furthermore, the functions $Q(z)$ and $f(z)$ from (17) have exponential growth
and the same zeros. Therefore,

$Q(z)=Ce^{\alpha z+i\beta z}f(z),\qquad
C\in{\mathbb{C}},\quad\alpha,\,\beta\in{\mathbb{R}}.$

Hence,

(33)
$\frac{Q^{\prime}(x+iy)}{Q(x+iy)}-\frac{f^{\prime}(x+iy)}{f(x+iy)}=\alpha+i\beta.$

It follows from (32) that the function $f^{\prime}(x+iy)/f(x+iy)$ holds (20)
for $y>s$. Let

$\log(1+H(x+iy))=\sum_{\gamma\in\Gamma_{1}}c_{\gamma}e^{2\pi i\gamma x},\quad
c_{\gamma}=c_{\gamma}(y),\quad\Gamma_{1}\subset(0,+\infty),\quad\sum_{\gamma\in\Gamma_{1}}|c_{\gamma}|<\infty.$

By (23),

$Q^{\prime}(x+iy)/Q(x+iy)=(\log
Q(x+iy))^{\prime}=\sum_{\gamma\in\Gamma_{1}}2\pi i\gamma c_{\gamma}e^{2\pi
i\gamma x}+2\pi i\kappa.$

Therefore, (33) and (20) yield

$\alpha=0,\quad\beta=\pi i(D+2\kappa),\quad 2\pi i\gamma c_{\gamma}=-2\pi
ie^{-2\pi\gamma y}b_{\gamma},\quad\Gamma_{1}=\Gamma\cap(0,+\infty),$

and

$\sum_{0<\gamma<1}|b_{\gamma}/\gamma|\leq e^{2\pi
y}\sum_{0<\gamma<1}|c_{\gamma}|<\infty.$

We obtain (8).

By (24), $1/Q(x\pm i\eta)\in\mathcal{H}$ for every $\eta>0$. The spectrum
$\Omega$ is bounded, hence $Q^{\prime}(x\pm i\eta)\in\mathcal{H}$ as well.
Therefore, $Q^{\prime}(x\pm i\eta)/Q(x\pm i\eta)\in\mathcal{H}$, and we get
the representation

$Q^{\prime}(x\pm i\eta)/Q(x\pm i\eta)=\sum_{\gamma\in
S}h_{\gamma}^{\pm}e^{2\pi i\gamma x},\qquad\sum_{\gamma\in
S}|h_{\gamma}^{\pm}|<\infty,$

where $S$ is a countable subsets of ${\mathbb{R}}$. Arguing as above, instead
of (31) we obtain the representation

$\hat{\mu}_{A}=\sum_{\gamma\in
S}\frac{h_{\gamma}^{-}e^{-2\pi\gamma\eta}-h_{\gamma}^{+}e^{2\pi\gamma\eta}}{2\pi
i}\delta_{\gamma}.$

Therefore, (32) is valid for all $s=\eta>0$. This gives (9).

I thank Professor Szilard Revesz from Renyi Institute of Mathematics for the
hospitality and useful discussions.

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# Rocket Landing Control with Grid Fins and Path-following using MPC

Junhao Yu, Jiarun (Sammy) Wei1 1Couse Project for ME 231A. {junhaoyu,
sammywei} @berkeley.edu

###### Abstract

This report is a summary of University of California, Berkeley’s ME C231A
final project. In this project, we attempt to optimize a landing trajectory of
a rocket. The goal is to minimize the total fuel consumption during the
landing process using different techniques. Once the optimal and feasible
trajectory is generated using batch approach, we attempt to follow the path
using a Model Predictive Control (MPC) based algorithm, called Trajectory
Optimizing Path-following Estimation from Demonstration (TOPED), in order to
generalize to similar initial states and models, where we introduce a novel
cost function for the MPC to solve. We further show that TOPED can follow a
demonstration trajectory well in practice under model mismatch and different
initial states.

##  I Problem Statement

Given a rocket dynamic system $f$ with states
$m,x,y,\dot{x},\dot{y},\theta,\dot{\theta}$ and inputs $F,\delta$: $m$ is
total mass of the rocket, $x$ is horizontal position of the rocket, $y$ is
vertical position of the rocket, $\dot{x}$ is horizontal velocity of the
rocket, $\dot{y}$ is the vertical velocity of the rocket, $\theta$ is the
angle between the center line of the rocket and the gravity line,
$\dot{\theta}$ the rate of change of $\theta$, $F$ is magnitude of the thrust
force, $\delta$ is the angle between the thrust force and the center line of
the rocket, $K$ is the ratio of $F$ and the rate of mass change $\dot{m}$, $J$
is moment of inertia of the rocket, and $l$ is length of the rocket, we wish
to minimize the cost of fuel during the landing process of the rocket given
the number of time steps. This objective can be formulated as an optimization
problem, and solved using batch method. Once the optimization is solved, we
obtain a planned optimal trajectory $\tau^{*}$ that realizes our objective.
Then we plan to use an MPC controller $\mathcal{C}$ to follow this planned
trajectory as closely as possible. We define a cost function for MPC which is
a weighted sum of the original cost function, L2 norm of states mismatch, L2
norm of input mismatch, and slack variables.

According to the above configuration, the state space representation and
dynamic model of the system are as follows:

$z(k)=\begin{bmatrix}m(k)&x(k)&y(k)&\dot{x}(k)&\dot{y}(k)&\theta(k)&\dot{\theta}(k)\end{bmatrix}^{T}$

The system dynamics for the original system without the grid fins are defined
as:

$z(k+1)=z(k)+T_{s}\cdot\begin{bmatrix}-\frac{F}{K}\\\ \dot{x}(k)\\\
\dot{y}(k)\\\ -\frac{F\sin(\theta+\delta)}{m}\\\
g-\frac{F\cos(\theta+\delta)}{m}\\\ \dot{\theta}(k)\\\
-\frac{Fl\sin(\delta)}{(2J)}\end{bmatrix}$

with the control input: $u=\left[F\;\delta\right]^{T}$.

The system dynamics for the new system with grid fins are defined as:

$z(k+1)=z(k)+T_{s}\cdot\begin{bmatrix}-\frac{F}{K}\\\ \dot{x}(k)\\\
\dot{y}(k)\\\ -\frac{F\cos(\theta+\delta)}{m}-\frac{\gamma\rho
A_{g}\dot{x}(k)^{2}}{2m}\\\ g-\frac{F\cos(\theta+\delta)}{m}-\frac{\gamma\rho
A_{g}\dot{y}(k)^{2}}{2m}\\\ \dot{\theta}(k)\\\
-\frac{Fl\sin(\delta)}{(2J)}\end{bmatrix}$

with control input: $u=\left[F\;\delta\;\gamma\right]^{T}$, where constant
$A_{g}$ denotes the maximum area of the grid fins and $\gamma$ is a grid fins
angle coefficient such that $0.8\leq\gamma\leq 1$.

##  II Related Work

Our project is highly related to ideas about optimal control and path
following techniques and learning from demonstration [1, 2, 3, 4, 5]. In [6],
the authors demonstrated an MPC-based approach to follow a precomputed
trajectory (path) in the setting of rocket landing guidance. Generally,
learning from demonstration is beneficial in scenarios when the demo model is
similar to the actual model of the system [7, 8, 9, 10]. Therefore, the agent
can learn from demo in order to generalize to slightly different models and
initial states. Recent literature has shown a variety of methods to learn from
demonstration [11, 12]. For example Dataset Aggregation (DAgger) method [13,
14, 15, 16, 17, 18] requires human labeling of a dataset and aggregates the
labeled dataset to the agent training data in order to converge to optimal
trajectory. Given a feasible trajectory, the agent can follow the trajectory
in a safe manner by simultaneously learning the safe set of MPC, as proposed
in [19].

##  III Methods and Experiments

###  III-A Optimal Landing Trajectory with Grid Fins

Consider the above dynamics models, the original system only has thrust vector
control while the new system has both TVC and grid fins. By adding grid fins,
the system gained extra lift and torque and thus outperformed original system
with merely Thrust Vector Control. We then solve for optimal trajectory using
fmincon. Based on our prediction, the rocket with grid fins would have
different trajectory due to the extra lift and torque, resulting in both less
fuel burned and better altitude angle ($\theta$) control.

###  III-B MPC Path-following

We propose a path-following algorithm, Trajectory Optimizaing Path-following
Estimation from Demonstrations (TOPED). Given a demonstration trajectory that
is feasible and optimal, we want to follow the trajectory $\tau^{*}$ using MPC
even under slightly imperfect and uncertain model and different initial
states. In TOPED, we require three different weight vectors, each representing
how important a state or input is in our path-following setting. The weights
can therefore be tuned to follow different states of interest. We also require
the system dynamics $f^{\prime}$ as the constraints in MPC’s receding horizon
optimal control problem. The optional slack variables $\lambda$ with weights
$\epsilon$ are used when the receding horizon control problem is infeasible.
In such cases, we add slackness to inputs, and solve the MPC again. Inspired
by [6], we define cost function at time step $k$ as $Q(x_{k},u_{k})=\lVert
W_{x}^{T}(x_{k}-x^{*}_{k})\rVert_{2}^{2}+\lVert
W_{u}^{T}(u_{k}-u^{*}_{k})\rVert_{2}^{2}+W_{J}J(x,u)^{2}$. The receding
horizon optimal control problem that Algorithm 1 is trying to solve is defined
as:

$\displaystyle\min\;{\sum_{k=0}^{N-1}Q(x_{k},u_{k})+\epsilon^{T}\lambda}$ (1)
$\displaystyle\textrm{s.t. }\;x(k+1)=f^{\prime}(x(k),u(k))\;\forall{k}$
$\displaystyle x_{k}\in\mathcal{X},u_{k}\in\mathcal{U}\;\forall{k}$
$\displaystyle x_{0}=x(0)$

where $x^{*}$ and $u^{*}$ are reference states and actions from demo. System
dynamics model $f^{\prime}$ could potentially be different than the original
model $f$ used in generating the optimal path by solving the optimal control
problem.

Algorithm 1 Trajectory Optimizing Path-following Estimation from
Demonstrations (TOPED)

0: Feasible demonstration path $\tau^{*}$; system model $f$; cost function
$J$; simulation horizon $M$; weight factors $W_{x}$, $W_{u}$, $W_{J}$; MPC
prediction horizon $N$, slack variables $\lambda$; slack variable weights
$\epsilon$

1: for $t$ $\in\\{1,...,M\\}$ do

2: Extract states and inputs from demo trajectory $\tau^{*}$ as $x^{*}$ and
$u^{*}$

3: Define loss function $Q(x,u)=\lVert
W_{x}^{T}(x_{t}-x_{t}^{*})\rVert_{2}^{2}+\lVert
W_{u}^{T}(u_{t}-u_{t}^{*})\rVert_{2}^{2}+W_{J}J(x,u)^{2}+\epsilon^{T}\lambda$

4: Run MPC with prediction horizon $N$ to minimize $Q(x,u)$, initially without
slack variables and their weights, $\epsilon^{T}\lambda=0$, obtain
$x^{opt}_{t}$ and $u^{opt}_{t}$, using $f$ as dynamics constraint

5: if INFEASIBLE then

6: Add slackness to input constraints, and increase weights of slack variables
as defined in $Q(x,u)$ and solve again until feasible

7: $x_{opt}\leftarrow x^{opt}_{t}$, $u_{opt}\leftarrow u^{opt}_{t}$

8: return optimal path-following trajectory $x_{opt},u_{opt}$ as
$\tau^{return}$

Given a precomputed (demonstration) optimal landing trajectory $\tau^{*}$, we
use an MPC-based Algorithm 1 to follow the trajectory as closely as possible.
We introduce a Gaussian noise $\mathcal{N}$ to our model in order to add
uncertainty to our model to make path-following more challenging.
Specifically, we add noise to some of the states and inputs readings, making
the model less perfect. Moreover, we start from three different initial
states: we now start with different amount of fuel and a location either
farther or nearer than the provided trajectory’s initial state. Since in our
scenario, we care about the horizontal and vertical locations of the rocket
the most, we assign more weight to those two states. Therefore, the second and
the third entries in $W_{x}$ should be significantly larger than others.
Intuitively, we assign more weight to the states and inputs we wish to follow,
and assign less weight to the less essential states and inputs. In our
setting, we want to follow states $x$ and $y$, inputs $F$ and $\delta$ as
closely as possible, so the weight vector we used was
$W_{x}=\left[1,5,5,0.5,1,0.1,0.5\right]$ and $W_{u}=\left[2,2,0.5\right]$. In
practice, we made the weight assigned on the original cost function $W_{J}=2$.
In the MPC controller of TOPED, we used a prediction horizon $N=4$, and our
simulation length is $M=20$ because the provided descent trajectory lands to
origin in 20 steps. In practice, we also found that when the initial states
are hugely different than the provided initial state (e.g. $x$ a lot greater
or smaller than 165), the problem becomes infeasible no matter how much
slackness we add to the input variables. Therefore, we ignore such cases and
only apply feasible initial states because the idea of path-following is to
generalize to similar situations. We also do not enforce the rocket to land
exactly at the origin. Instead, we add slackness to the final location of the
rocket so that it is within 15 meters from the origin.

##  IV Results

###  IV-A Optimal Descent Path Planning

We plot comparison of landing trajectory, $\theta$, thrust command and fuel-
burned percentage between the original system and the system enhanced with
grid fins. The grid fin command is plotted as well. The plots are shown in Fig
1. Though the two paths (a) have similar ‘S’ shape, we can tell the difference
through altitude angle (b), fuel-consumption curve (c) and thrust command (d):
The system with grid fins stabilized $\theta$ faster without any oscillation
while the original system experienced an oscillation. With the extra lift
provided by grid fins, the new system tends to glide rather than landing
directly, which saves around 8 percent of carried fuel.

|
---|---
|
|

(a) $x$ and $y$ path

(b) Altitude angle $\theta$

(c) Fuel Consumption Percentage

(d) Thrust changes

(e) Grid fin coefficient

Figure 1: States and input changes in optimal landing trajectory for each
model. Top row: $x$ and $y$ path during landing, and angle $\theta$ changes.
Middle row: Fuel consumption changes and thrust changes. Bottom row: Grid fin
coefficient changes.

###  IV-B MPC Path-following

Our MPC path-following using Algorithm 1 can follow the trajectory closely. To
illustrate here, in Fig. 2, we plot out how TOPED follows the precomputed
horizontal and vertical location $(x,y)$ of the rocket trajectory since our
interest is to follow the descent path closely. The final locations under
three different initial states of the rocket are within 10 meters from the
origin. It can be observed that the MPC controller is trying to follow the
demonstration path closely under slightly different models and initial states.

Figure 2: Path following trajectory under model mismatch and different initial
state. Initial condition 1 is 15 meters farther horizontally and 100 meters
farther vertically, with 55kg less fuel to begin with. Initial condition 2 is
10 meters farther horizontally and 100 meters nearer vertically with 100kg
less fuel to begin with. Initial condition 3 is 10 meters nearer horizontally
and 100 meters farther vertically, with the same amount of fuel. All three
test models are injected with random Gaussian noise $\mathcal{N}$.

One advantage of MPC is that in our experiment, the optimal reference
demonstration path was generated in roughly 30 minutes on a PC with NVIDIA
GTX1060. Since the MPC problem is shortsighted, unlike batch approach, we only
look ahead 4 steps while closely following the demonstration trajectory, so
the MPC path-following problem was solved in only about 56.75 seconds, which
is significantly faster than batch approach, and the final trajectory is near
optimal.

##  V Conclusion

In this projects we explored some techniques used to optimize fuel consumption
in rocket landing. Meanwhile, we proposed an MPC-based algorithm that can be
used to follow a provided trajectory, and we experimentally showed that using
Algorithm 1, the runtime complexity to solve the optimization problem
decreases, while maintaining the near-optimality of the trajectory.

Since our system is non-linear, we do not attempt to make any convergence
guarantee about TOPED, although we can say with similar initial states and
models, Algorithm 1 will drive the controller to follow the demonstration
path. This notion of similarity can be quantitatively measured using KL-
divergence if we introduce random variable noise to the model. Discussion of
KL-divergence is beyond the scope of this report, but readers are welcomed to
contact us for more information.

In terms of future work, we would explore our ideas in a scenario of three
dimensional model, because our experiment was done in a 2D setting, which is a
lot easier. Furthermore, since the optimal path was solved using optimization
techniques, we would like to see if a human manipulated simulation dataset can
be used to provide demonstration and reference. In this way, we no longer need
to solve the optimization problem, and with such a dataset, one can simulate
and optimize the landing and explore better trajectories using reinforcement
learning techniques, and transfer to real environments, which can generalize
to significantly more scenarios, not just limited to similar models, goals,
and initial states.

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