Large Fourier Quasicrystals and Wiener's Theorem

TL;DR

This paper introduces new conditions for when a discrete measure's support is a finite union of translated lattices, using Wiener's Theorem and almost periodic functions to advance mathematical understanding.

Contribution

It provides novel simple criteria linking measure support structures to harmonic analysis tools like Wiener's Theorem.

Findings

Support of discrete measures characterized as finite unions of translated lattices.

New conditions derived from Wiener's Theorem and almost periodic functions.

Enhanced understanding of measure support structures in Euclidean space.

Abstract

We find new simple conditions for support of a discrete measure on Euclidean space to be a finite union of translated lattices. The arguments are based on a local analog of Wiener's Theorem on absolutely convergent trigonometric series and theory of almost periodic functions.

Equations

$$p_m(\phi )=\underset{{\mathbb{R}}^d}{\sup }(\max \{1,|x|\}{)}^m\underset{|k_1|+\cdots +|k_d|\le m}{\max }|D^k(\phi (x))|,m=0,1,2,\dots ,$$

$$|f(\phi )|\le C{p}_m(\phi ).$$

$$\hat{f}(\phi )=f(\hat{\phi })\text{mbox}forall\phi \in S({\mathbb{R}}^d),$$

$$\hat{\phi }(y)={\int }_{{\mathbb{R}}^d}\phi (x)\exp \{-2\pi i⟨x,y⟩\}d{m}_d(x)$$

$$\mu =\sum _{\lambda \in \Lambda }{a}_{\lambda }{\delta }_{\lambda },\hat{\mu }=\sum _{\gamma \in \Gamma }{b}_{\gamma }{\delta }_{\gamma },\log \left[\sum _{|\lambda |<r}|a_{\lambda }|+\sum _{|\gamma |<r}|b_{\gamma }|\right]=O(\log r),r\to \infty ,$$

$$|\hat{\mu }|(B(r))=O(r^d)\text{mbox}asr\to \infty .$$

$$\{\tau \in {\mathbb{R}}^d:\underset{x\in {\mathbb{R}}^d}{\sup }|f(x+\tau )-f(x)|<\epsilon \}$$

$$\mathcal{M}(f)=\underset{r\to \infty }{\lim }\frac{1}{{\omega }_d{r}^d}{\int }_{B(x,r)}f(y)d{m}_d(y),$$

$$a_f(\gamma )=\mathcal{M}\left(f(x)e^{-2\pi i⟨x,\gamma ⟩}\right).$$

$$(\hat{f},\psi )=(f,\hat{\psi })=\sum _n{c}_n{\int }_{{\mathbb{R}}^d}{e}^{2\pi i⟨y,{\gamma }_n⟩}\hat{\psi }(y)dy=\sum _n{c}_n\psi ({\gamma }_n).$$

$$(\psi \star \mu )(s)=\int \psi (s-x)d\mu (x)=\int \hat{\psi }(y)e^{2\pi i⟨y,s⟩}d\hat{\mu }(y)=\sum _{\gamma \in \Gamma }b(\gamma )\hat{\psi }(\gamma )e^{2\pi i⟨s,\gamma ⟩}.$$

$$\int |\hat{\psi }(y)|d|\hat{\mu }|(y)\le {\int }_0^{\infty }C\min \{1,t^{-N}\}dM(t)=C\underset{t\to \infty }{\lim }\frac{M(t)}{t^N}+CN{\int }_1^{\infty }\frac{M(t)}{t^{N+1}}dt,$$

$$\underset{R\to \infty }{\lim }\frac{1}{{\omega }_d{R}^d}{\int }_{B(R)}{e}^{-2\pi i⟨x,\lambda ⟩}d\mu (x)$$

$$\frac{1}{R^d}{\int }_{{\mathbb{R}}^d}\psi (|x|/R)e^{-2\pi i⟨x,\lambda ⟩}d\mu (x)=\{R^{-d}\psi (|x/R|)\mu (x)\}\hat{}(\lambda )=\hat{\psi }(R(\lambda -y))\star \hat{\mu }(y),$$

$$\hat{\psi }(R(\lambda -y))\star \hat{\mu }(y)={\int }_{|\lambda -y|<1}\hat{\psi }(R(\lambda -y))d\hat{\mu }(y)+{\int }_{|\lambda -y|\ge 1}\hat{\psi }(R(\lambda -y))d\hat{\mu }(y).$$

$$c_N{\int }_{|\lambda -y|\ge 1}(R|y-\lambda |{)}^{-N}|\hat{\mu }|(y)=c_N{R}^{-N}{\int }_1^{\infty }{t}^{-N}d|\hat{\mu }|(B(\lambda ,t)).$$

$$\left|{\int }_{{\mathbb{R}}^d}\psi (|x|/R)e^{-2\pi i⟨\lambda ,x⟩}d\mu (x)-{\omega }_d^{-1}{\int }_{B(x,R)}{e}^{-2\pi i⟨\lambda ,x⟩}d\mu (x)\right|\le C\epsilon R^d.$$

$$\eta =(1/2)\underset{\lambda ,{\lambda }^{\prime }\in \Lambda ,\lambda \ne {\lambda }^{\prime }}{\inf }|\lambda -{\lambda }^{\prime }|,\beta =\underset{\lambda \in \Lambda }{\inf }|a_{\lambda }|.$$

$$\left|\sum _{\lambda \in \Lambda }{a}_{\lambda }\psi (x-\lambda )-\sum _{{\lambda }^{\prime }\in \Lambda }{a}_{{\lambda }^{\prime }}\psi (x+\tau -{\lambda }^{\prime })\right|<\beta .$$

$$\left|\sum _{\lambda \in \Lambda }\phi (x-\lambda )-\sum _{{\lambda }^{\prime }\in \Lambda }\phi (x+\tau -{\lambda }^{\prime })\right|<\epsilon .$$

$$\underset{\lambda \in \Lambda }{\sup }|a_{\lambda }|\le \underset{x\in {\mathbb{R}}^d}{\sup }\left|\int \psi (|x-\lambda |)d\mu (\lambda )\right|=\underset{x\in {\mathbb{R}}^d}{\sup }\left|{\int }_{{\mathbb{R}}^d}\hat{\psi }(y)e^{2\pi i⟨x,y⟩}d\hat{\mu }(y)\right|\le {\int }_{{\mathbb{R}}^d}|\hat{\psi }(y)|d|\hat{\mu }|(y).$$

$$F(\theta ,\tau )=\sum _{n\in {\mathbb{Z}}^N}{c}_n(\tau )e^{2\pi i⟨\theta ,n⟩}$$

$$c_n(\tau )={\int }_{[0,1{]}^N}F(\theta ,\tau )e^{-2\pi i⟨\theta ,n⟩}d\theta .$$

$$c_n(\tau )=\frac{-1}{4{\pi }^2{n}_1^2}{\int }_{[0,1{]}^N}\frac{{\partial }^{2}F(\theta ,\tau )}{\partial {\theta }_1^2}{e}^{-2\pi i⟨\theta ,n⟩}d\theta .$$

$$c_n(\tau )=\frac{(-1{)}^m}{(2\pi {)}^{2m}{n}_1^2\cdots n_m^2}{\int }_{[0,1{]}^N}\frac{{\partial }^{2m}F(\theta ,\tau )}{\partial {\theta }_1^2\dots \partial {\theta }_m^2}{e}^{-2\pi i⟨\theta ,n⟩}d\theta .$$

$$|c_n(\tau )|\le \frac{1}{(2\pi {)}^{2m}{n}_1^2\cdots n_m^2}\underset{\theta ,\tau }{\max }\left|\frac{{\partial }^{2m}F(\theta ,\tau )}{\partial {\theta }_1^2\dots \partial {\theta }_m^2}\right|.$$

$$|c_n(\tau )|\le C(F)\min \{1,n_1^{-2}\}\cdots \min \{1,n_N^{-2}\}.$$

$$H(z)={\int }_{{\mathbb{R}}^d}h(z-\zeta )\phi (|\zeta |)d{m}_2(\zeta )$$

$$F(\theta ,\tau )=H\left(\sum _{n\le N}{c}_n{e}^{2\pi i{\theta }_n}+2\eta e^{2\pi i\tau }\right)$$

$$H(S(x)+2\eta e^{2\pi i\tau })=\sum _n{c}_n(\tau )e^{2\pi i⟨x,{\rho }_n⟩}\in W,{\rho }_n\in {\mathrm{Lin}}_{\mathbb{Z}}\{{\lambda }_1,\dots ,{\lambda }_N\},$$

Full text

Large Fourier Quasicrystals and Wiener’s Theorem

S.Yu.Favorov

Sergii Favorov,

iii Karazin’s Kharkiv National University

iii Svobody sq., 4,

iii 61022, Kharkiv, Ukraine

sfavorov@gmail.com

Abstract. We find new simple conditions for support of a discrete measure on Euclidean space to be a finite union of translated lattices. The arguments are based on a local analog of Wiener’s Theorem on absolutely convergent trigonometric series and theory of almost periodic functions.

AMS Mathematics Subject Classification: 52C23, 42B10

Keywords: distribution, Fourier transform, measure with discrete support, spectrum of measure, almost periodic measure, lattice, Weiner’s Theorem

1. introduction

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

[TABLE]

$k=(k_{1},\dots,k_{d})\in({\mathbb{N}}\cup\{0\})^{d},\ D^{k}=\partial^{k_{1}}_{x_{1}}\dots\partial^{k_{d}}_{x_{d}}$. These norms generate a topology on $S({\mathbb{R}}^{d})$, and elements of the space $S^{\prime}({\mathbb{R}}^{d})$ of continuous linear functionals on $S({\mathbb{R}}^{d})$ are called tempered distributions. For every tempered distribution $f$ there exist $C>0$ and $m\in{\mathbb{N}}\cup\{0\}$ such that for all $\varphi\in S({\mathbb{R}}^{d})$

[TABLE]

Moreover, this estimate is sufficient for distribution $f$ to be in $S^{\prime}({\mathbb{R}}^{d})$ (see [16], Ch.3).

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

[TABLE]

where

[TABLE]

is the Fourier transform of the function $\varphi$. Note that the Fourier transform of every tempered distribution is also a tempered distribution. Here we consider only the case when $f$ and $\hat{f}$ are measures on ${\mathbb{R}}^{d}$.

To formulate the results of the paper we need some notions and definitions.

Set $B(x,r)=\{y\in{\mathbb{R}}^{d}:\,|y-x|<r\}$, $B(r)=B(0,r)$. We say that $E\subset{\mathbb{R}}^{d}$ is a full-rank lattice if $E=T{\mathbb{Z}}^{d}$ for some nondegenerate linear operator $T$ on $R^{d}$; $E$ is relatively dense if there exists $R<\infty$ such that $B(x,R)\cap E\neq\emptyset$ for each $x\in{\mathbb{R}}^{d}$; $E$ is discrete if $E\cap B(x,1)$ is finite for all $x\in{\mathbb{R}}^{d}$; $E$ is uniformly discrete if $|x-x^{\prime}|\geq\varepsilon>0$ for all $x,\,x^{\prime}\in E,x\neq x^{\prime}$.

A Radon measure $\mu$ is discrete (uniformly discrete) if its support is discrete (uniformly discrete), $\mu$ is translation bounded if its variations $|\mu|$ are uniformly bounded on balls of radius 1, and $\mu$ is slowly increasing if $|\mu|(B(r))$ grows at most polynomially as $r\to\infty$. Note that every translation bounded measure is slowly increasing, and every slowly increasing measure belongs to $S^{\prime}({\mathbb{R}}^{d})$.

Following [9] we say that $\mu$ is a Fourier quasicrystal if both measures $\mu$, $\hat{\mu}$ are slowly increasing atomic measures. More precisely, we will suppose that

[TABLE]

where $\delta_{z}$ means the unit mass at the point $z$ and the sets $\Lambda,\,\Gamma$ are countable. In the paper we usually consider the case of discrete support $\Lambda$ and countable spectrum $\Gamma$ of $\mu$. We say that $\mu$ is a large Fourier quasicrystal if, in addition, $\inf_{\lambda\in\Lambda}|a_{\lambda}|>0$. The simplest example of a Fourier quasicrystal is the measure $\mu_{0}=\sum_{k\in{\mathbb{Z}}^{d}}\delta_{k}$. By the Poisson formula we have $\hat{\mu}_{0}=\mu_{0}$.

Such measures are the main object in the theory of Fourier quasicrystals (see [2]-[12]). The corresponding notions were inspired by experimental discovery of non-periodic atomic structures with diffraction patterns consisting of spots, which was made in the mid ’80s.

In the paper we are interested in the cases when support $\Lambda$ is a subset of a finite union of translated full-rank lattices. Several very interesting results of this type were obtained by N.Lev, A.Olevskii in [8], [9]. Here it is one of them.

Theorem 1** ([9]).**

Let $\mu$ be a Fourier quasicrystal on ${\mathbb{R}}^{d}$ with discrete spectrum $\Gamma$ and uniformly discrete set of differences $\Lambda-\Lambda$. Then $\Lambda$ is a subset of a finite union of translates of a single full-rank lattice $L$, and $\Gamma$ is a subset of a finite union of translates of the conjugate lattice.

Also, there exists a Fourier quasicrystal  with countable spectrum $\Gamma$ such that $\Lambda-\Lambda$ is uniformly discrete, but $\Lambda$ is not contained in a finite union of translates of any lattice.

We prove the following theorem, which amplifies the previous one

Theorem 2**.**

Let $\mu$ be a large Fourier quasicrystal on ${\mathbb{R}}^{d}$ with a discrete set of differences $\Lambda-\Lambda$. Then $\Lambda$ is a finite union of translates of a single full-rank lattice $L$.

In the next section we will give a generalization of Theorem 2 for pairs of Fourier quasicrystals.

Results of another type of results were obtained by Y.Meyer and A.Cordoba.

Theorem 3** (Y.Meyer [11]).**

Let $\mu=\sum_{\lambda\in\Lambda}a_{\lambda}\delta_{\lambda}$ be a measure with discrete support $\Lambda$ and $a_{\lambda}\in S$ for some finite set $S\subset{\mathbb{C}}\setminus\{0\}$. If $\mu\in S^{\prime}({\mathbb{R}})$ and if its Fourier transform $\hat{\mu}$ is a translation bounded measure on ${\mathbb{R}}$, then $\Lambda=E\triangle\bigcup_{j=1}^{N}(\alpha_{j}{\mathbb{Z}}+\beta_{j})$, where $\alpha_{j}>0,\ \beta_{j}\in{\mathbb{R}}$, and the set $E$ is finite.

Here $A\triangle B$ means the symmetrical difference between $A$ and $B$.

In [5] M.Kolountzakis extended the above theorem to measures on ${\mathbb{R}}^{d}$. He replaced the condition ”the measure $\hat{\mu}$ is translation bounded” with the weaker one

[TABLE]

He also found a condition for support $\mu$ to be a finite union of several full-rank lattices. His result is very close to Cordoba’s one:

Theorem 4** ([2]).**

Let $\mu$ be a uniformly discrete Fourier quasicrystal on ${\mathbb{R}}^{d}$ with $a_{\lambda}$ belonging to a finite set $F$. If the measure $\hat{\mu}$ is translation bounded, then $\Lambda$ is a finite union of translates of several, possibly incommensurable, full-rank lattices.

In paper [4] we replaced the conditions ”$a_{\lambda}$ from a finite set” by ”$|a_{\lambda}|$ from a finite set” and the condition ”$\hat{\mu}$ is translation bounded” by (1).

Now we obtain the following theorem.

Theorem 5**.**

Let $\mu$ be a uniformly discrete large Fourier quasicrystal on ${\mathbb{R}}^{d}$, and let $\hat{\mu}$ satisfy (1). Then $\Lambda$ is a finite union of translates of several disjoint full-rank lattices.

The proof is based on an analog of Wiener’s Theorem on Fourier series:

Theorem 6** (N.Wiener, see, for example, [17], Ch.VI).**

Let $F(t)=\sum_{n\in{\mathbb{Z}}}c_{n}e^{2\pi int}$ be an absolutely convergent Fourier series, and $h(z)$ be a holomorphic function on a neighborhood of the set $\overline{\{F(t):\,t\in[0,1]\}}$. Then the function $h(F(t))$ admits an absolutely convergent Fourier series expansion as well.

Denote by $W$ the class of absolutely convergent series $\sum_{n}c_{n}e^{2\pi i\langle x,\gamma_{n}\rangle},\ \gamma_{n}\in{\mathbb{R}}^{d}$. We prove the following theorem

Theorem 7**.**

Let $K\subset{\mathbb{C}}$ be an arbitrary compact, $h(z)$ be a holomorphic function on a neighborhood of $K$, and $f\in W$. Then there is a function $g\in W$ such that if $f(x)\in K$ then $g(x)=h(f(x))$.

If $K=\overline{f({\mathbb{R}}^{d})}$ then we obtain the global Wiener’s Theorem for almost periodic functions.

Finally, we prove an analog of Wiener’s Theorem for Fourier quasicrystals:

Theorem 8**.**

Let $\mu=\sum_{\lambda\in\Lambda}a_{\lambda}\delta_{\lambda}$ be a uniformly discrete Fourier quasicrystal on ${\mathbb{R}}^{d}$ with the translation bounded measure $\hat{\mu}$. Suppose that $h(z)$ is a holomorphic function on a neighborhood of the closure of the set ${\{a_{\lambda}:\,\lambda\in\Lambda\}}$. Then $\nu=\sum_{\lambda\in\Lambda}a_{\lambda}h(a_{\lambda})\delta_{\lambda}$ is also Fourier quasicrystal with translation bounded $\hat{\nu}$.

Corollary**.**

Let $\mu=\sum_{\lambda\in\Lambda}a_{\lambda}\delta_{\lambda}$ be a uniformly discrete Fourier quasicrystal on ${\mathbb{R}}^{d}$ with translation bounded measure $\hat{\mu}$. Let $A$ be a connected component of the set $\overline{{\{a_{\lambda}:\,\lambda\in\Lambda\}}}$ such that $0\notin A$. Then $\{\lambda\in\Lambda:\,a_{\lambda}\in A\}$ is a finite union of translates of several full-rank lattices.

2. Large quasicrystals and almost periodic measures

We recall some definitions related to the notion of almost periodicity (see, for example, [10]).

A continuous function $f$ on ${\mathbb{R}}^{d}$ is almost periodic, if for every $\varepsilon>0$ the set of $\varepsilon$-almost periods of $f$

[TABLE]

is a relatively dense set in ${\mathbb{R}}^{d}$.

For every almost periodic function $f$ there exists the limit uniform in $x\in{\mathbb{R}}^{d}$

[TABLE]

where $\omega_{d}$ is volume of the unit ball. The Fourier coefficient of $f$ at frequency $\gamma\in{\mathbb{R}}^{d}$ is

[TABLE]

For every $f$ only countably many $a_{f}(\gamma)$ do not vanish, and the set $\{a_{f}(\gamma)\}$ defines the function $f$ uniquely.

It can be easily proved that every function $f=\sum_{n}c_{n}e^{2\pi i\langle x,\gamma_{n}\rangle}\in W$ is almost periodic. We set $\|f\|_{W}=\sum_{n}|c_{n}|$.

Since ${\mathcal{M}}\left(e^{2\pi i\langle x,\gamma\rangle}e^{2\pi i\langle x,-\gamma^{\prime}\rangle}\right)=\begin{cases}1,&\gamma=\gamma^{\prime},\\ 0,&\gamma\neq\gamma^{\prime},\end{cases}$  we get $a_{f}(\gamma)=\begin{cases}c_{n},&\gamma=\gamma_{n},\\ 0,&\gamma\neq\gamma_{n}.\end{cases}$

Moreover, $\hat{f}=\sum_{n}c_{n}\delta_{\gamma_{n}}$. Indeed, if $\psi\in S({\mathbb{R}}^{d})$, then

[TABLE]

Next, a (complex) measure $\mu$ on ${\mathbb{R}}^{d}$ is almost periodic, if for every continuous function $\psi$ on ${\mathbb{R}}^{d}$ with compact support the function $(\psi\star\mu)(x)$ is almost periodic in $x\in{\mathbb{R}}^{d}$. It is clear that support of every almost periodic measure is relatively dense in ${\mathbb{R}}^{d}$.

A discrete set $\Lambda$ is almost periodic, if measure $\sum_{\lambda\in\Lambda}\delta_{\lambda}$ is almost periodic.

Theorem 9** (L.Ronkin [14]).**

Every almost periodic measure is translation bounded.

A connection between almost periodicity of measure and properties of its Fourier transform was obtained by Y.Meyer:

Theorem 10** ([10]).**

Let $\mu$ and its Fourier transform $\hat{\mu}$ be translation bounded measures. Then $\mu$ is almost periodic iff spectrum of $\mu$ is countable.

Here we need an extension of this result.

Theorem 11**.**

Let $\mu$ be a measure from $S^{\prime}({\mathbb{R}}^{d})$, and let $\hat{\mu}$ be a slowly increasing measure. Then $\mu$ is almost periodic iff it is translation bounded and has a countable spectrum.

First we prove some lemmas.

Lemma 1**.**

Let $\mu$ be a measure from $S^{\prime}({\mathbb{R}}^{d})$ with countable spectrum, let $\hat{\mu}$ be a slowly increasing measure, and let $\psi\in S({\mathbb{R}}^{d})$. Then convolution $\psi\star\mu$ belongs to $W$.

Proof. Let $\hat{\mu}=\sum_{\gamma\in\Gamma}b_{\gamma}\delta_{\gamma}$. We have

[TABLE]

Since $\hat{\psi}\in S({\mathbb{R}}^{d})$, we get

[TABLE]

where $M(t)=|\hat{\mu}|(B(t))$. For appropriate $N$ the latter integral is finite. Therefore the sum in (2) converges absolutely.

Lemma 2**.**

Let $\mu$ be a translation bounded measure, and let $\hat{\mu}$ be a slowly increasing measure. Then for each $\lambda\in{\mathbb{R}}^{d}$ the limit

[TABLE]

exists and equals $\hat{\mu}(\{\lambda\})$.

Proof. Pick an arbitrary $\varepsilon>0$. Let $\psi(|x|)$ be $C^{\infty}$-differentiable function on ${\mathbb{R}}^{d}$ such that $\psi(t)=1/\omega_{d}$ for $t<1-\varepsilon$, $\psi(t)=0$ for $t>1+\varepsilon$, $0\leq\psi(t)\leq 1/\omega_{d}$ for $1-\varepsilon\leq t\leq 1+\varepsilon$, and $\int_{{\mathbb{R}}^{d}}\psi(|x|)dx=1$. We have

[TABLE]

then

[TABLE]

Note that $\hat{\psi}(0)=1$ and $\hat{\psi}(R(\lambda-y))\to 0$ for $\lambda\neq y$ as $R\to\infty$. By the Dominated Convergence Theorem, the first integral in (3) tends to [math] if $\hat{\mu}(\{\lambda\})=0$ and to $\hat{\mu}(\{\lambda\})$ otherwise. Since $\hat{\psi}\in S({\mathbb{R}}^{d})$, we see that the second integral in (3) does not exceed

[TABLE]

If $N$ is large enough, then $|\hat{\mu}|(B(\lambda,t))=O(t^{N-1})$ as $t\to\infty$. Arguing as in the previous lemma, we obtain that the second integral in (3) tends to [math] as $R\to\infty$.

Furthermore, since the measure $\mu$ is translation bounded, we get for large $R$

[TABLE]

Since $\varepsilon$ is arbitrary, we get the assertion of the Lemma.

Remark. We used property of $\mu$ to be translation bounded only in the latter part of the proof.

Proof of Theorem 11. Suppose that $\mu$ is translation bounded measure with countable spectrum and slowly increasing measure $\hat{\mu}$. By Lemma 1, $\psi\star\mu(s)$ is almost periodic for every $\psi\in S({\mathbb{R}}^{d})$. Check that $(\phi\star\mu)(t)$ is almost periodic for each continuous function $\phi$ with a compact support in the ball $B(R)$. Let $\psi_{n}\in S(R^{d}),\ \operatorname{supp}\psi_{n}\subset B(R+1)$, be a sequence that uniformly converges to $\phi$. Since $\mu$ is translation bounded, the almost periodic functions $(\psi_{n}\star\mu)(t)$ uniformly converge to $(\phi\star\mu)(t)$, therefore the latter function is almost periodic, and $\mu$ is an almost periodic measure.

Now suppose that $\mu$ is almost periodic. By Theorem 9, $\mu$ is translation bounded. Let $\nu$ be the atomic component of measure $\hat{\mu}$, let $\mu_{1}$ be the inverse Fourier transform of $\nu$, and $\mu_{2}$ be the inverse Fourier transform of $\hat{\mu}-\nu$. It follows from the first part of the proof that $\mu_{1}$ is an almost periodic measure. Therefore, $\mu_{2}$ is almost periodic too. Let $\varphi$ be $C^{\infty}$-function with compact support. The convolution $\varphi\star\mu_{2}$ is an almost periodic function, hence measure $\varphi\star\mu_{2}(x)m_{d}(x)$ is translation bounded. By Lemma 2, Fourier coefficient of $\varphi\star\mu_{2}$ at each frequency $\lambda\in{\mathbb{R}}^{d}$ equals $(\varphi\star\mu_{2})\hat{\,}(\{\lambda\})=\hat{\varphi}(\lambda)\hat{\mu}_{2}(\{\lambda\})=0$. Therefore, $\varphi\star\mu_{2}\equiv 0$ for each $\varphi\in S({\mathbb{R}}^{d})$ with compact support. Then $\mu_{2}=0$ and $\hat{\mu}=\nu$.

To prove Theorem 2 we need two lemmas.

Lemma 3**.**

Let $\mu=\sum_{\lambda\in\Lambda}a_{\lambda}\delta_{\lambda},\ |a_{\lambda}|\geq\varepsilon>0$, be an almost periodic measure on ${\mathbb{R}}^{d}$ with uniformly discrete $\Lambda$. Then the set $\Lambda$ is almost periodic too.

Proof. Check that for every continuous function $\varphi(x)$ with compact support and every $\varepsilon>0$ the function $\sum_{\lambda\in\Lambda}\varphi(x-\lambda)$ has a relatively dense set of $\varepsilon$-almost periods. Set

[TABLE]

Without loss of generality suppose that $\operatorname{supp}\varphi\subset B(\eta)$, and $0\leq\varphi(x)\leq\varphi(0)=1$. There exists $\rho\in(0,\eta)$ such that the inequality $|x-x^{\prime}|<\rho$ implies $|\varphi(x)-\varphi(x^{\prime})|<\varepsilon$. Set $\psi(x)=\varphi(\eta x/\rho)$. Note that if $\psi(x)\neq 0$, then $|x|<\rho$. Let $\tau$ be $\beta$-almost period of the function $\psi\star\mu(x)=\sum_{\lambda\in\Lambda}a_{\lambda}\psi(x-\lambda)$. We have for all $x\in{\mathbb{R}}^{d}$

[TABLE]

Clearly, at most one term in every sum does not vanish. For $x=\lambda$ we get $|a_{\lambda}-a_{\lambda^{\prime}}\psi(\lambda+\tau-\lambda^{\prime})|<\beta$, hence $\psi(\lambda+\tau-\lambda^{\prime})\neq 0$ and $|\lambda+\tau-\lambda^{\prime}|<\rho$. Therefore, $|\varphi(x-\lambda)-\varphi(x+\tau-\lambda^{\prime})|<\varepsilon$. If there is another $\lambda^{\prime\prime}\in\Lambda$ such that $\varphi(x+\tau-\lambda^{\prime\prime})\neq 0$, then $|x+\tau-\lambda^{\prime\prime}|<\eta$ and $|\lambda^{\prime}-\lambda^{\prime\prime}|<2\eta$. This is impossible, therefore for all $x\in{\mathbb{R}}^{d}$

[TABLE]

Since there is a relatively dense set of $\beta$-almost periods $\tau$ of the function $\psi\star\mu$, we conclude that the function $\sum_{\lambda\in\Lambda}\varphi(x-\lambda)$ is almost periodic too.

Lemma 4**.**

Let $\mu=\sum_{\lambda\in\Lambda}a_{\lambda}\delta_{\lambda}$ be a uniformly discrete measure from $S^{\prime}({\mathbb{R}}^{d})$ with the slowly increasing measure $\hat{\mu}$. Then $\sup_{\lambda\in\Lambda}|a_{\lambda}|<\infty$ and, consequently, the measure $\mu$ is translating bounded.

Proof. Set $\eta=(1/2)\inf_{\lambda,\lambda^{\prime}\in\Lambda,\lambda\neq\lambda^{\prime}}|\lambda-\lambda^{\prime}|$. Let $\psi(|y|)$ be a $C^{\infty}$-function such that $\operatorname{supp}\psi(|y|)\subset B(\eta)$ and $\psi(0)=1$. We have

[TABLE]

Taking into account that $\hat{\psi}(y)\in S(R^{d})$ and arguing as above in Lemma 1, we obtain that the latter integral is finite.

Our proof of Theorem 2 is based on the following result.

Theorem 12** ([3]).**

Let $\Lambda$ be an almost periodic set in ${\mathbb{R}}^{d}$ with a discrete set of differences $\Lambda-\Lambda$. Then $\Lambda$ is a finite union of translates of a single full-rank lattice.

Proof of Theorem 2. Since the set $\Lambda-\Lambda$ is discrete, we get that $\Lambda$ is uniformly discrete. By Lemma 4, $\mu$ is translation bounded. By Theorem 11, $\mu$ is almost periodic. Lemma 3 implies that $\Lambda$ is almost periodic as well. Now, using Theorem 12, we obtain the assertion of Theorem 2.

There is an analog of Theorem 12 for pairs of measures.

Theorem 13**.**

Let $\mu_{1},\,\mu_{2}$ be large Fourier quasicrystals such that the set $\Lambda_{1}-\Lambda_{2}$ of differences of their supports $\Lambda_{1},\,\Lambda_{2}$ is discrete. Then there exists a full-rank lattice $L$ such that the both supports $\Lambda_{1},\,\Lambda_{2}$ are finite unions of translates of $L$.

In fact, the proof of this theorem is the same as in the previous one. But instead of Theorem 12 we need to use the corresponding assertion for pairs of measures from [4].

3. Wiener’s Theorem for quasicrystals

In this section we prove Theorems 7 and 8. We begin with the following lemma.

Lemma 5**.**

Let $F(\theta,\tau),\ \theta=(\theta_{1},\dots,\theta_{N})\in[0,1]^{N},\,\tau\in[0,1]$, be $C^{\infty}$-differentiable function in all variables and periodic with period $1$ in each $\theta_{j},\ j=1,\dots,N$. Then its Fourier series

[TABLE]

converges absolutely and uniformly in $\tau\in[0,1]$.

Proof. We have

[TABLE]

Let $n=(n_{1},\dots,n_{N})$ with $n_{1}\neq 0$. Integrating by parts in variable $\theta_{1}$ two times, we get

[TABLE]

If $n_{1}\cdot n_{2}\cdots n_{m}\neq 0$, then after integrating by parts in variables $\theta_{1},\dots,\theta_{m}$ we get

[TABLE]

Therefore,

[TABLE]

Consequently, for all $n\in{\mathbb{Z}}^{N}$ we get

[TABLE]

Obvious convergence of the series $\sum_{n\in{\mathbb{Z}}^{k}}\min\{1,n_{1}^{-2}\}\cdots\min\{1,n_{N}^{-2}\}$ implies the assertion of the Lemma.

Proof of Theorem 7. Let $U$ be a neighborhood of $K$ such that the function $h(z)$ is holomorphic and bounded on $U$. Set $\eta=(1/4)\operatorname{dist}(K,{\mathbb{C}}\setminus U)$. Let $\varphi(|z|)$ be $C^{\infty}$-differentiable nonnegative function with support in $B(\eta)$ such that $\int_{B(\eta)}\varphi(|z|)dm_{2}(z)=1$. Then the function

[TABLE]

is $C^{\infty}$-differentiable on ${\mathbb{R}}^{d}$ and coincides with $h(z)$ on the set $\{z:\operatorname{dist}(z,K)\leq 3\eta\}$.

Suppose $f(x)=\sum_{n}c_{n}e^{2\pi i\langle x,\gamma_{n}\rangle}$. Choose $N<\infty$ such that for $S(x)=\sum_{n\leq N}c_{n}e^{2\pi i\langle x,\gamma_{n}\rangle}$ we get $\|f(x)-S(x)\|_{W}<\eta$. Applying Lemma 5 to the function

[TABLE]

and replacing $\theta_{j}$ with $\langle x,\gamma_{j}\rangle,\ j=1,\dots,N$, we get

[TABLE]

and $\|H(S(x)+2\eta e^{2\pi i\tau})\|_{W}\leq C(N)$ uniformly in $\tau\in[0,1]$.

Next, taking into account that $|f(x)-S(x)|\leq\|f-S\|_{W}<\eta$, we have

[TABLE]

Clearly, this sum belongs to $W$ and its norm is uniformly bounded in $\tau\in[0,1]$. Hence the same assertion is valid for the product of (4) and (5). Consequently, the function

[TABLE]

belongs to $W$.

If $f(x)\in K$, then $\operatorname{dist}(S(x),K)<\eta$ and $B(S(x),2\eta)\subset\{\zeta:\operatorname{dist}(\zeta,K)<3\eta\}$. Moreover, $f(x)\in B(S(x),2\eta)$ and the function $H(z)$ coincides with $h(z)$ for $z\in B(S(x),2\eta)$. Therefore, $g(x)=h(f(x))$ in this case.

To deduce Theorem 8 from Theorem 7 we prove the following two lemmas.

Lemma 6**.**

Let $\mu$ be a slowly increasing measure with a translation bounded measure $\hat{\mu}$. Then for all $f\in W$ the Fourier transform $(f\mu)\hat{\ }$ is a translation bounded measure.

Proof. It is evident that $f\mu$ is a slowly increasing measure. Set $\mu_{\gamma}(x)=e^{2\pi i\langle x,\gamma\rangle}\mu(x)$ for $\gamma\in{\mathbb{R}}^{d}$. Clearly, for each $y\in{\mathbb{R}}^{d}$ we get $|\hat{\mu}_{\gamma}|(B(y,1))=|\hat{\mu}|(B(y-\gamma,1))$. Suppose $f(x)=\sum_{n}c_{n}e^{2\pi i\langle x,\gamma_{n}\rangle}\in W$. Then $(f\mu)(x)=\sum_{n}c_{n}\mu_{\gamma_{n}}(x)$ and $(f\mu)\hat{\ }=\sum_{n}c_{n}\widehat{\mu}_{\gamma_{n}}(x)$. Therefore for each $y\in{\mathbb{R}}^{d}$

[TABLE]

Lemma 7**.**

Let $\mu=\sum_{\lambda\in\Lambda}a_{\lambda}\delta_{\lambda}$ be a uniformly discrete measure with countable spectrum and slowly increasing Fourier transform. Then there is a function $f\in W$ such that $f(\lambda)=a_{\lambda}$ for all $\lambda\in\Lambda$.

Proof. Set $\eta=(1/2)\inf_{\lambda,\lambda^{\prime}\in\Lambda,\lambda\neq\lambda^{\prime}}|\lambda-\lambda^{\prime}|$. Let $\psi$ be $C^{\infty}$-differentiable function with support in $B(\eta)$ such that $\psi(0)=1$. Then the function $f(x)=\psi\star\mu(x)=\sum_{\lambda\in\Lambda}a_{\lambda}\psi(x-\lambda)$ satisfies the condition $f(\lambda)=a_{\lambda}$ for all $\lambda\in\Lambda$. By Lemma 1, $f\in W$.

Proof of Theorem 8. We apply Theorem 7 with $K=\overline{\{a_{\lambda}:\,\lambda\in\Lambda\}}$ and $f\in W$ such that $f(\lambda)=a_{\lambda}$, which exists by Lemma 7, and then we apply Lemma 6 for the measure $\mu$ and the function $g$ from Theorem 7.

4. Generalization of Meyer’s Theorem

To prove the results of this section we need some definitions.

Lattice is a discrete subgroup of $R^{d}$. If $A$ is a lattice or a coset of some lattice in ${\mathbb{R}}^{d}$, then $\operatorname{dim}A$ is the dimension of the smallest translated subspace of ${\mathbb{R}}^{d}$ that contains $A$. Every lattice $L$ of dimension $k$ has the form $T{\mathbb{Z}}^{k}$, where $T:\,{\mathbb{Z}}^{k}\to{\mathbb{Z}}^{d}$ is a linear operator of rank $k$. For $k=d$ we get a full-rank lattice. Also, the coset ring of an abelian topological group $G$ is the smallest collection of subsets of $G$, that is closed under finite unions, finite intersections and complements and which contains all cosets of all open subgroups of $G$. Next, Bohr compactification $\mathfrak{R}$ of ${\mathbb{R}}^{d}$ is a compact group, its dual is ${\mathbb{R}}^{d}$ with the discrete topology, ${\mathbb{R}}^{d}$ is a dense subset of $\mathfrak{R}$ with respect to the topology on $\mathfrak{R}$, and restrictions to ${\mathbb{R}}^{d}$ of continuous functions on $\mathfrak{R}$ are just almost periodic functions on ${\mathbb{R}}^{d}$, in particular, they are bounded and continuous on ${\mathbb{R}}^{d}$ (see for example [13]).

Both Meyer’s Theorem 3 and our Theorem 5 are based on Cohen’s Idempotent Theorem:

Theorem 14** ([1]).**

Let $G$ be a locally compact abelian group and $\hat{G}$ its dual group. If $\nu$ is a finite Borel measure on $G$ and is such that its Fourier transform $\hat{\nu}(\lambda)\in\{0,1\}$ for all $\lambda\in\hat{G}$, then the set $\{\lambda:\,\hat{\nu}(\lambda)=1\}$ is in the coset ring of $\hat{G}$.

We will apply Theorem 14 to Bohr compactification $\mathfrak{R}$ of ${\mathbb{R}}^{d}$ and its dual ${\mathbb{R}}^{d}$ with the discrete topology. Then we will use the following theorem by M.Kolountzakis:

Theorem 15** ([5]).**

Elements of the ring of cosets of ${\mathbb{R}}^{d}$ in the discrete topology, which are discrete in the usual topology of ${\mathbb{R}}^{d}$, are precisely finite unions of sets of the type

[TABLE]

Proof of Theorem 5. Let $\rho$ be the inverse Fourier transform of the measure $\mu$, let $\psi(|x|)\in S({\mathbb{R}}^{d})$ be the function with compact support from Lemma 2 with $\varepsilon=1/2$.

Set $\rho_{R}(y)=R^{-d}\psi(|y/R|)\rho(y)$. Integrating by parts and using (1), we get

[TABLE]

Therefore,

[TABLE]

Arguing as above in Lemma 2, we get that $\hat{\rho}_{R}(x)$ tends to $\hat{\rho}(\{\lambda\})=\mu(\{\lambda\})=a_{\lambda}$ for $x=\lambda\in\Lambda$ and to zero for $x\notin\Lambda$.

Since variations of the measures $\rho_{R}$ are uniformly bounded, they act on all bounded functions on ${\mathbb{R}}^{d}$, and hence also on all functions from $C(\mathfrak{R})$. Therefore there exists a measure $\mathfrak{r}$ on $\mathfrak{R}$ with a finite total variation $|\mathfrak{r}|$, and a subsequence $R^{\prime}$ such that $\rho_{R^{\prime}}\to\mathfrak{r}$ in the weak–star topology. In other words, $\langle\rho_{R^{\prime}},f\rangle\to\langle\mathfrak{r},f\rangle$ as $R^{\prime}\to\infty$ for all $f\in C(\mathfrak{R})$. Applying this to every character of $\mathfrak{R}$ in place of $f$ we obtain

[TABLE]

Note that $\hat{\mathfrak{r}}(x)$ is a continuous function with respect to the discrete topology on ${\mathbb{R}}^{d}$, and $|a_{\lambda}|\leq|\mathfrak{r}|$ for all $a_{\lambda}$.

Using Lemma 7, choose a function $f\in W$ such that $f(\lambda)=a_{\lambda}$ for all $\lambda\in\Lambda$. Taking into account that $\inf_{\lambda}|a_{\lambda}|>0$ and using Theorem 7 with $h(z)=1/z$, we construct the function $g(x)=\sum_{n}c_{n}e^{2\pi i\langle x,\tau_{n}\rangle}\in W$ with the property $g(\lambda)=1/a_{\lambda}$. Since $\hat{\mathfrak{r}}(y+\tau)=e^{2\pi i\langle x,\tau\rangle}\hat{\mathfrak{r}}$, we see that the Fourier transform of a finite sum $\sum_{n\leq N}c_{n}\mathfrak{r}(y+\tau_{n})$ is equal to $\sum_{n\leq N}c_{n}e^{2\pi i\langle x,\tau_{n}\rangle}\hat{\mathfrak{r}}$. The variation of the measure $\sum_{n\leq N}c_{n}\mathfrak{r}(y+\tau_{n})$ does not exceed $\sum_{n\leq N}|c_{n}||\mathfrak{r}|(\mathfrak{R})$, therefore the previous assertion is valid for infinite sums. Hence the Fourier transform of the measure $\mathfrak{b}=\sum_{n}c_{n}\mathfrak{r}(y+\tau_{n})$ equals $1$ for $x=\lambda\in\Lambda$ and [math] for $x\not\in\Lambda$. By Theorem 14, $\Lambda$ is in the coset ring of ${\mathbb{R}}^{d}$ in discrete topology. Taking into account Lemma 4 and Theorem 11, we see that the measure $\mu$ is almost periodic. Therefore, by Lemma 3, measure $\sum_{\lambda\in\Lambda}\delta_{\lambda}$ is also almost periodic.

Using (6) and the notion of almost periodicity, it is easy to get the assertion of our theorem. By definition, put $\delta_{A}=\sum_{x\in A}\delta_{x}$ for a countable set $A\subset{\mathbb{R}}^{d}$. We have

[TABLE]

where $A_{k},\,B_{j},\,C_{i}$ are cosets such that $\dim A_{k}=d,\,\dim B_{j}<d,\,\dim C_{i}<d$. Let $A_{1}=x_{1}+L_{1},\ A_{2}=x_{2}+L_{2}$ be two cosets of dimension $d$. If $\dim A_{1}\cap A_{2}=d$, then $A_{1}\cap({\mathbb{R}}^{d}\setminus A_{2})$ is a finite union of disjoint cosets of $L_{1}\cap L_{2}$, and the same is valid for $A_{2}\cap({\mathbb{R}}^{d}\setminus A_{1})$. Therefore we can replace $A_{1}\cup A_{2}$ by a finite sum of disjoint cosets $A^{\prime}_{l},\,\dim A^{\prime}_{l}=d$, hence $\delta_{A_{1}\cup A_{2}}=\sum_{l}\delta_{A^{\prime}_{l}}$. For every two cosets $H_{1},\,H_{2}$ with $H_{1}\cap H_{2}\neq\emptyset$ and $\dim(H_{1}\cap H_{2})<d$ we get $\delta_{H_{1}\cup H_{2}}=\delta_{H_{1}}+\delta_{H_{2}}-\delta_{H_{1}\cap H_{2}}$. Repeating this transformation for each pair of cosets with nonempty intersection, we obtain a representation of $\delta_{\Lambda}$ after a finite number of steps of the form

[TABLE]

where $D_{s},\,E_{s},\,F_{s}$ cosets, $\dim D_{s}=d$, $\dim E_{s}<d$, and $\dim F_{s}<d$. Obviously, every coset $D_{s}$ has $d$ linearly independent periods, hence $\delta_{D_{s}}$ is an almost periodic measure, and so is $\sum_{s=1}^{N_{1}}\delta_{D_{s}}$. Therefore the measure

[TABLE]

is almost periodic too. But its support is contained in a finite union of hyperplanes and isn’t relatively dense. Therefore the measure (7) is identically zero, and $\delta_{\Lambda}=\sum_{s=1}^{N_{1}}\delta_{D_{s}}$. Since the measure $\delta_{\Lambda}$ has the unit masses at every point of $\Lambda$, we see that the cosets $D_{s}$ are disjoint and $\Lambda=\cup_{s=1}^{N_{1}}D_{s}$.

Proof of the Corollary. By Lemma 4, $A$ is a bounded set. Apply Theorem 8 with a function $h(z)$ that equals $1$ on a neighborhood $U$ of $A$ and [math] on ${\mathbb{C}}\setminus U$, and then Theorem 5 for $\sum_{\lambda\in\Lambda,a_{\lambda}\in A}a_{\lambda}\delta_{\lambda}$.