1 Introduction

Random perturbations of discrete structures have been studied by many authors from different viewpoints. For instance, a problem of interest in statistical physics is to study what happens with the statistical properties (e.g, two-point correlation, structure factor) of the perturbed discrete sets; we refer to [10, 16] and the references therein. Another problem of interest in Diffraction theory is to compare the diffraction measure of a random perturbation with the diffraction measure of the deterministic discrete set which is perturbed; see for instance [2, Chapter 11] and [13] for more details. Very recently, in [24] Yakir deals with random perturbations of a lattice where it is shown that, under some assumptions on the random displacements, the lattice can be recovered almost surely from the perturbations. In this work we focus on random perturbations and reconstructibility in quasi-periodic structures, including quasicrystals, and on weakening the independence conditions on the random perturbations. We stress that quasicrystals exist not only as mathematical objects, but also as naturally ocurring [6] physical materials, which although rare, have been argued to be ubiquitous throughout the universe [5]. Since the discovery of quasicrystalline structures in materials by Shechtman and his collaborators [21] many efforts have been dedicated to understanding their underlying order and their stability under perturbations (see for instance [13, 14, 19] just to mention a few).

In more precise terms, if \(X\subset {\mathbb {R}}^d\) is a quasi-periodic set, we let \((\xi _p)_{p\in X}\) be a sequence of identically distributed random vectors defined in a probability space \((\Omega , {\mathcal {F}},{\mathbb {P}})\), with common distribution \(\xi \). A random perturbation of X is a realization of the random set

$$\begin{aligned} X_{\xi }:=\{p+\xi _p(\omega ):\ p\in X\}. \end{aligned}$$

For two random processes \(X,Y\subset {\mathbb {R}}^d\) that take values into locally finite configurations, we say that X is almost surely recovered from Y if there exists a measurable function F defined over the essential range of the process Y such that almost surely \(F(Y)=X\); this notion extends naturally to random measures. If X is a fixed (deterministic) subset of \({\mathbb {R}}^d\), we identify X with the point process which takes constant value equal to X.

In this work we focus on finding conditions under which the Fourier transform of \(X_{\xi }\) determines the Fourier transform of a quasi-periodic set X almost surely, based on an explicit formula that links these two Fourier transforms. It follows that X can then be almost surely recovered from \(X_\xi \), but we can say even more. In fact, some mathematical and physical point configurations can be classified in terms of their hidden order (e.g, repetitivity of separated nets [17], quasi-periodicity [2, 15], Hyperuniformity [19, 23]). In this context, our main results (Theorems A,B and C below) provide another way to distinguish the large class of point sets that have diffraction measure singular with respect to Lebesgue measure (see hypotheses (H1), (H2) in Sect. 1) in terms of the robustness of their Fourier Transform. This allows to recognize these configurations under random perturbations, by taking a Fourier Transform. Note that as a special case our setting includes quasicrystals (defined by asking that their diffraction measure is purely atomic), which in turn include the classical case of lattices, for which the diffraction measure is supported on their dual lattice. In [1] the authors show that SCD patterns have singular diffraction measure (in particular they satisfy our hypotheses), but are not quasicrystals.

Notations We include here some basic definitions and notations which we will use throughout this work. We use the notation \(|\cdot |\) indistinguishably for the euclidean norm in \({\mathbb {R}}^d\) and for the module (resp. absolute value) of a complex number (resp. real number), which will be clear in each context. As usual we denote by \(B(x,R):=\{y\in {\mathbb {R}}^d: |x-y|<R\}\) the euclidean open ball centered at \(x\in {\mathbb {R}}^d\) with radius \(r>0\) and we use the notation \(B_R:=B(0,R)\).

We denote by Vol the Lebesgue measure in \({\mathbb {R}}^d\) and we denote the cardinality of a finite set \(A\subset {\mathbb {R}}^d\) by \(\# A\). Observe that a locally integrable function \(f\in L_{loc}^1({\mathbb {R}}^d)\) can be identified with \(f\cdot Vol\); see Sect. 3 for more details. Furthermore, for an atomic measure \(\mu \) we write \(\mu (p)\) instead of \(\mu (\{p\})\). Finally, for strictly positive real-valued functions f, g we use the standard Landau asymptotic notations O(f(R)) and o(g(R)).

2 Hypotheses on the Deterministic Set X

Throughout this work we assume that X is an uniformly discrete set, which means that there exists \(r>0\) such that \(||x-y||\ge r\) for every \(x,y\in X\). The set X is assumed to satisfy the following two conditions:

  1. (H.1)

    There exists a positive constant \(\textrm{dens}(X)\), called the asymptotic density of X, such that for every \(R>0\) we have

    $$\begin{aligned} |\# (X\cap B_R)-\textrm{dens}(X)\cdot Vol(B_R)|=o(R^d). \end{aligned}$$
  2. (H.2)

    The diffraction measure \(\widehat{\gamma _X}\) of \(\delta _X:=\sum _{p\in X}\delta _p\), given by

    $$\begin{aligned} \forall f\in C^\infty _c({\mathbb {R}}^d),\qquad \langle \widehat{\gamma _X}, f\rangle :=\lim _{R\rightarrow \infty }\frac{1}{Vol(B_R)}\int _{{\mathbb {R}}^d} f(\lambda )\left| \sum _{p\in X\cap B_R} e^{-2\pi i\langle p, \lambda \rangle }\right| ^2 d\lambda \nonumber \\ \end{aligned}$$
    (1.1)

    exists and is singular with respect to Lebesgue measure, where \(\delta _p\) denotes the Dirac mass at the point \(p\in X\).

There are plenty of uniformly discrete sets X satisfying the hypothesis (H.1) (where trivially all lattices are included). Non-trivial examples are cut-and-project sets constructed with an appropriate choice of the window; see Sect. 4.4.2 for precise definitions. A particular case are the cut-and-project sets which are bounded displacement equivalent to a lattice; these sets include, for instance, the set of vertices of the Penrose tilling. It is worth mentioning that a generic cut-and-project set in \({\mathbb {R}}^d\) is, up to bounded displacement, equivalent to a lattice (for more details see [12]). Another interesting example of a set X satisfying (H.1) is the \(X=V_d\), known as the set of visible points of \({\mathbb {Z}}^d\), and defined as \(V_d:=\{x\in {\mathbb {Z}}^d:\ gcd(x)=1\}\), where gcd(x) denotes the greatest common divisor of the coordinates of \(x\in {\mathbb {Z}}^d\); it is known that \(\textsf{dens}(V_d)=1/\zeta (d)\), where \(\zeta \) denotes the Riemann’s zeta function. We refer to [4] for the proof of (H.1) in this case. Observe that \(V_d\) is not relatively dense, which roughly speaking amounts to saying that \(V_d\) has arbitrarily large holes.

All the abovementioned examples satisfy (H.2). See for instance Corollary 9.3 and Proposition 9.9 in [2] and [2, Section 9.4] for several examples of cut-and-project sets and the precise computations of their diffraction measures; see also [4] for a proof that \(\widehat{\gamma _{V_d}}\) is purely atomic in the case of visible points of a lattice. In particular a set X satisfies (H.2) if its weak dual \(X^*\) is discrete, where \(X^*\) is the support of the Fourier transform of \(\delta _X:=\sum _{p\in X}\delta _p\), given by \(\widehat{\delta _X}=\sum _{\lambda \in X^*}A(\lambda )\delta _\lambda \). More explicitly, there exist \(\{A(\lambda )\}_{\lambda \in X^*}\) such that

$$\begin{aligned} \displaystyle \lim _{R\rightarrow \infty }\frac{1}{Vol(B_R)}\sum _{p\in X\cap B_R}e^{-2\pi i\langle p,\lambda \rangle }=\left\{ \begin{array}{ll} A(\lambda ) &{}\text { for }\lambda \in X^{*},\\ 0&{}\text { else.} \end{array}\right. \end{aligned}$$

Note that \(X^*\) coincides with the usual dual for lattices and cut-and-project sets.

We consider the general setting given by (H.2) since there are uniformly discrete point configurations in \({\mathbb {R}}^d\) for which only the absolutely continuous part of their diffraction spectrum vanishes; see for instance [1].

3 Statements of the Results

Recall that \(X_{\xi }\) is the realization random set \(\{p+\xi _p:\ p\in X\}\). For every \(\lambda \in {\mathbb {R}}^d\), \(\omega \in \Omega \) and \(R>0\), we define \(M_{R,\xi ,\omega }(\lambda )\) by

$$\begin{aligned} M_{R,\xi ,\omega }(\lambda ):=\dfrac{1}{Vol(B_R)}\sum _{p\in X_{\xi }\cap B_R}e^{-2\pi i\langle p,\lambda \rangle }. \end{aligned}$$

The aim of this work is to show that under our hypotheses (H.1) and (H.2) the Fourier transform of X is preserved under random perturbations. As a byproduct, this allows us to recover almost surely the quasi-periodic set X from \(X_{\xi }\).

Theorem A

Suppose \(X\subset {\mathbb {R}}^d\) satisfies hypotheses (H.1) and (H.2) from Sect. 1. Assume that the \(\xi _p\)’s are i.i.d and there exists a positive number \(\varepsilon \) such that \({\mathbb {E}}(|\xi |^{d+\varepsilon })<\infty \). Then, almost surely the following holds for every \(\lambda \in {\mathbb {R}}^d\),in the sense of vague convergence of measures:

$$\begin{aligned} \displaystyle \lim _{R\rightarrow \infty }M_{R,\xi ,\omega }(\lambda )={\mathbb {E}}\left[ e^{-2\pi i\langle \xi ,\lambda \rangle }\right] \widehat{\delta _X}(\lambda ), \end{aligned}$$
(2.1)

where \(\widehat{\delta _X}\) is the (weak-)Fourier transform of X given by \(\lim _{R\rightarrow \infty }\frac{\widehat{\delta _{X\cap B_R}}}{Vol(B_R)}\).

A key step in the proof of Theorem A is the strong law of large numbers. Since the hypothesis of this well-known result can be weakened, we can adapt our proof to show that the conclusions of Theorem A still hold for non-independent identically distributed random vectors under mild assumptions on the correlations (see Sect. 4.2 for the precise definitions).

Theorem B

Assume that X satisfies (H.1) and (H.2), that the \(\xi (p), p\in X\) are identically distributed, and that there is \(\varepsilon >0\) such that \({\mathbb {E}}(|\xi |^{d+\varepsilon })<\infty \). If \((\xi _p)_{p\in X}\) is a compactly-mixing random process such that for \(Y_p:=|\xi _p|\mathbbm {1}_{|\xi _p|\le |p|}\) there holds

$$\begin{aligned} \sum _{N\ge 1}\frac{1}{N^2}\sum _{\begin{array}{c} p,q\in B_N\\ p\ne q \end{array}}|\textsf{Cov}(Y_p,Y_q)|<\infty , \end{aligned}$$
(2.2)

then almost surely the limit (2.1) still holds for every \(\lambda \in {\mathbb {R}}^d\).

We note that a stronger version of Theorems A, B is achievable if we assume that X is an asymptotically affine perturbation of a lattice. In this case we only need the \(\xi _p\) to be weakly mixing. See Theorem D and Sect. 4.3 for details.

Though the conclusion of Theorem B holds for every uniformly discrete set X satisfying hypotheses (H.1) and (H.2), we require strong assumptions on the correlations. However, if the random process \((\xi _p)_{p\in X}\) is such that the random perturbation \(X_{\xi }\) is stationary, we have the same conclusion of Theorem B but under more classical conditions on the correlations, namely weak-mixing.

Theorem C

Assume that X satisfy the hypotheses (H.1) and (H.2) and that \((\xi _p)_{p\in X}\) is a process of random vectors for which there is a positive number \(\varepsilon \) such that \({\mathbb {E}}[|\xi |^{d+\varepsilon }]<\infty \). If the random perturbation \(X_{\xi }\) is stationary under the action of \({\mathbb {R}}^d\) on itself by translations and weakly-mixing, then almost surely, the limit 2.1 holds for every \(\lambda \in {\mathbb {R}}^d\).

As we mentioned before, if the limit (2.1) holds and \({\mathbb {E}}(e^{-2\pi i\langle \xi ,\lambda \rangle })\ne 0\), this allow us to recover the Fourier transform \(\widehat{\delta _X}\) from \(\widehat{\delta _{X_{\xi }}}\) without requiring specific conditions on X. The above condition on \(\xi \) holds in several cases of interest, for example for \(\xi \) a mixture of Gaussians.

Corollary 2.1

Suppose that either the hypotheses of Theorem A, B or C hold. If for all \(\lambda \in {\mathbb {R}}^d\) there holds \({\mathbb {E}}(e^{-2\pi i\langle \xi ,\lambda \rangle })\ne 0\), then \(\widehat{\delta _{X}}\) is almost surely recoverable from \(\widehat{\delta _{X_{\xi }}}\). More precisely, almost surely, for all \(\lambda \in {\mathbb {R}}^d\) there holds

$$\begin{aligned} \widehat{\delta _X}(\lambda )={\mathbb {E}}\left( e^{-2\pi i\langle \xi ,\lambda \rangle }\right) ^{-1}\widehat{\delta _{X_{\xi }}}(\lambda ). \end{aligned}$$

A class of results closely related to our work concern the recovery of the structure factor \(S_X\) from the one random perturbations \(S_{X_\xi }\). In fact, [10, 16] obtain a formula for \(S_{X_{\xi }}\) in terms of the structure factor \(S_X\) of the unperturbed data X (here X itself is assumed to be a stationary random process). This yields a formula related to (2.1) but in expectation form. This relation is discussed in Sect. 5.2.

4 Basics on Mathematical Diffraction Theory

In this section we introduce some terminology and basic concepts which we use along this work. Let \({\mathcal {S}}({\mathbb {R}}^d)\) be the Schwartz space of rapidly decreasing \(C^{\infty }\) (complex-valued) test functions. The Fourier transform of \(f\in {\mathcal {S}}({\mathbb {R}}^d)\) is defined by

$$\begin{aligned} {\widehat{f}}(y):=\int f(x)e^{-2\pi i\langle x,y\rangle }dx. \end{aligned}$$

By a (complex) measure we mean a linear functional on the set of compactly supported functions \(C_c({\mathbb {R}}^d)\) such that for every compact set \(K\subset {\mathbb {R}}^d\) there is a positive constant \(M_K\) satisfying that

$$\begin{aligned} |\mu (f)|\le M_K||f||_{\infty }, \end{aligned}$$

for every \(f\in C_c({\mathbb {R}}^d)\) supported on K, and where \(||\cdot ||_{\infty }\) denotes the supremum norm on \(C_c({\mathbb {R}}^d)\). By the Riesz Representation Theorem, there is an equivalence between this definition of measure and the classical measure-theoretic concept of regular Radon measure. When a measure \(\mu \) determines a tempered distribution \(T_{\mu }(f)=\mu (f)\) (i.e a continuous linear functional defined on \({\mathcal {S}}({\mathbb {R}}^d)\)) via the formula

$$\begin{aligned} \forall f\in {\mathcal {S}}({\mathbb {R}}^d),\qquad T_{\mu }(f):=\int f(x)d\mu (x), \end{aligned}$$

then its Fourier transform is defined by \({\widehat{\mu }}(f):=\mu ({\widehat{f}})\).

The total variation \(|\mu |\) of a measure \(\mu \) is the smallest positive measure for which \(|\mu (f)|\le |\mu |(f)\) holds for every non-negative \(f\in C_c({\mathbb {R}}^d)\). A measure \(\mu \) is said to be translation bounded if for every compact \(K\subset {\mathbb {R}}^d\) there is a positive number \(a_K\) such that \(|\mu |(x+K)\le a_K\), for every \(x\in {\mathbb {R}}^d\).

Let \(P_{\mu }:=\{x\in {\mathbb {R}}^d:\ \mu (\{x\})\ne 0\}\) be the set of atoms of \(\mu \). We say that \(\mu \) is purely atomic (or pure point) if it has atoms only, i.e, \(\mu (A)=\sum _{x\in A\cap P_{\mu }}\mu (\{x\})\) for every Borel set A; moreover \(\mu \) is called continuous if \(\mu (\{x\})=0\) for every \(x\in {\mathbb {R}}^d\). A measure \(\mu \) is said to be absolutely continuous with respect to a measure \(\nu \), if there exists \(g\in L^{1}_{loc}(\nu )\) such that \(\mu =g\nu \). Furthermore \(\mu \) and \(\nu \) are called mutually singular (which is denoted by \(\mu \perp \nu \)) if there exists a relatively compact Borel set \(S\subset {\mathbb {R}}^d\) such that \(|\mu |(S)=0\) and \(|\nu |({\mathbb {R}}^d\setminus S)=0\) (with obvious interpretation for the value of \(|\mu |(A)\)).

For \(f\in {\mathcal {S}}({\mathbb {R}}^d)\), define \({\widetilde{f}}(x):=\overline{f(-x)}\), and for a measure \(\mu \) we consider \({\widetilde{\mu }}(f):=\overline{\mu ({\widetilde{f}})}\). We say that \(\mu \) is positive definite if \(\mu (f*{\widetilde{f}})\ge 0\) for every \(f\in C_c({\mathbb {R}}^d)\). By Proposition 8.6 in [2], every positive definite measure is Fourier transformable, and its Fourier transform is a positive translation bounded measure.

Recall that the convolution \(\mu *\nu \) between two measures \(\mu \) and \(\nu \) is defined by

$$\begin{aligned} \mu *\nu (f):=\displaystyle \int f(x+y)d\mu (x)d\nu (y); \end{aligned}$$

If \(\mu \) is translation bounded and \(\nu \) is finite, then the convolution \(\mu *\nu \) is well-defined and is also a translation bounded measure. Moreover, whenever \({\widehat{\mu }}\) is a measure, one has the identity \(\widehat{\mu *\nu }={\widehat{\mu }}{\widehat{\nu }}\) (see Theorem 8.5 in [2]).

We say that a net of measures \((\mu _R)_{R>0}\) (resp. positive definite measures) converges vaguely to a measure \(\mu \) if for every test function \(f\in C_c({\mathbb {R}}^d)\) (resp. for \(f\in {\mathcal {S}}({\mathbb {R}}^d)\))

$$\begin{aligned} \lim _{R\rightarrow \infty }\langle \mu _R,f\rangle =\langle \mu ,f\rangle , \end{aligned}$$

where we use the notation \(\langle \mu ,f \rangle :=\mu (f)\).

We denote by \(\mu _R\) the restriction of \(\mu \) over the ball \(B_R\). The autocorrelation measure of \(\mu \) is defined by

$$\begin{aligned} \gamma _{\mu }:=\lim _{R\rightarrow \infty }\dfrac{\mu _R*\widetilde{\mu _R}}{Vol(B_R)}, \end{aligned}$$

whenever this limit exists. In this case, \(\gamma _{\mu }\) is the limit of positive definite measures, thus is positive definite, and then it is Fourier transformable and its Fourier transform is positive definite and translation bounded. We also call \(\widehat{\gamma _{\mu }}\) the diffraction measure of \(\mu \).

If \(\mu :=\sum _{p\in X}\mu (p)\delta _p\), where \(\delta _p\) is the Dirac measure at \(p\in X\), to deal with the diffraction measure of \(\mu \) we need to prove that, in the sense of vague convergence of measures,

$$\begin{aligned} \lim _{R\rightarrow \infty } \frac{1}{Vol(B_R)}\left| \sum _{p\in X\cap B_R}\mu (p)e^{-2\pi i \langle p,y\rangle }\right| ^2 dy= \widehat{\gamma _{\mu }} \end{aligned}$$

In more abstract terms, we are required to prove the following:

$$\begin{aligned} \lim _{R\rightarrow \infty }{\mathcal {F}}\left( \frac{1}{Vol(B_R)} \mu _R*\widetilde{\mu _R}\right) ={\mathcal {F}}\left( \lim _{R\rightarrow \infty } \frac{1}{Vol(B_R)}\mu _R*\widetilde{\mu _R}\right) , \end{aligned}$$
(3.1)

where \({\mathcal {F}}\) denotes the Fourier transform. This follows from Lemma 4.11.10 in [3], which states that a net of positive definite measures converging weakly in the sense of measures then their Fourier transforms converge weakly as well.

5 Proofs

The proofs of Theorems A,B and C rely on Lemma 4.1 and Proposition 4.2 below. Lemma 4.1 is a slightly variation of Lemma 2 in [24], with the very same proof. It claims that the cardinality of points in \(X\cap B_R\) which escape from \(B_R\) under random perturbations converges in mean to 0 almost surely (the same happens for the cardinality of points in \(X\setminus B_R\) with perturbations lying inside to \(B_R\)).

Lemma 4.1

Assume that X satisfies hypothesis (H.1) and that there is \(\varepsilon >0\) such that \(\sup _{p\in X}{\mathbb {E}}(|\xi _p|^{d+\varepsilon })<\infty \). Then almost surely

$$\begin{aligned} \begin{aligned} \lim _{R\rightarrow \infty }\dfrac{\#\{p\in X:\ |p|\le R, |p+\xi _p(\omega )|>R\}}{Vol(B_R)}=0, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \lim _{R\rightarrow \infty }\dfrac{\#\{p\in X:\ |p|>R, |p+\xi _p(\omega )|\le R\}}{Vol(B_R)}=0 \end{aligned}$$

Proposition 4.2 below roughly speaking says that, under the hypotheses of Theorems A, B or C, the limit defining (2.1) holds when the indices in the sum from \(M_{R,\xi ,\omega }\) are taken over \(X\cap B_R\) instead of \(X_{\xi }\cap B_R\).

Proposition 4.2

Suppose that either the hypotheses of Theorem A, B or C hold. Then almost surely for every \(\lambda \in {\mathbb {R}}^d\) we have that

$$\begin{aligned} \lim _{R\rightarrow \infty }\frac{1}{Vol(B_R)}\sum _{p\in X\cap B_R}e^{-2\pi i\langle p, \lambda \rangle }\left( e^{-2\pi i\langle \xi _p,\lambda \rangle } - {\mathbb {E}}\left[ e^{-2\pi i \langle \xi ,\lambda \rangle }\right] \right) =0. \end{aligned}$$
(4.1)

As in [24], the conclusions of Theorems A, B and C follow from Lemma 4.1 and Proposition 4.2 since

$$\begin{aligned} \sup _{\lambda \in {\mathbb {R}}^d}\left| M_{R,\xi ,\omega }(\lambda )-\frac{1}{Vol(B_R)}\sum _{p\in X\cap B_R}e^{-2\pi i\langle p+\xi _p,\lambda \rangle }\right| \\ \le \dfrac{\#\{p\in X:|p|\le R,\ |p+\xi _p|>R\}}{Vol(B_R)}+\dfrac{\#\{p\in X:\ |p|> R,\ |p+\xi _p|\le R\}}{Vol(B_R)}\longrightarrow 0, \end{aligned}$$

where the last convergence holds almost surely by Lemma 4.1. Hence by the above limit and by Proposition 4.2 there holds (2.1).

We now direct our attention to proving Proposition 4.2 for X satisfying the hypotheses (H.1) and (H.2). To do this in the context of mathematical diffraction theory, we define

$$\begin{aligned} \mu _X^{\lambda }:=\sum _{p\in X}e^{-2\pi i \langle p, \lambda \rangle }\delta _p \end{aligned}$$
(4.2)

and

$$\begin{aligned} \mu _\xi ^{\lambda }:=\sum _{p\in X}\left( e^{-2\pi i\langle \xi _p,\lambda \rangle } - {\mathbb {E}}[e^{-2\pi i \langle \xi ,\lambda \rangle }]\right) \delta _p. \end{aligned}$$
(4.3)

By extending the strategy of [24], we will aim to show that the diffraction measures of \(\mu _X^{\lambda }\) and \(\mu _{\xi }^{\lambda }\), denoted by \(\widehat{\gamma _\xi ^\lambda }:=\widehat{\gamma _{\mu _\xi ^\lambda }}\) and \(\widehat{\gamma _X^\lambda }:=\widehat{\gamma _{\mu _X^\lambda }}\) respectively, exist and are almost surely mutually singular. In fact, we will prove in Lemma 4.3 that \(\widehat{\gamma _{X}^{\lambda }}\) is singular with respect to Lebesgue measure and in Propositions 4.4, 4.7 and 4.16 below that \(\widehat{\gamma _{\xi }^{\lambda }}\) is almost surely absolutely continuous with respect to the Lebesgue measure, under the hypotheses of Theorems A,B and C respectively. Thus the fact that \(\widehat{\gamma _X^\lambda }\perp \widehat{\gamma _\xi ^\lambda }\) is then used in combination with Theorem A.4 in order to show (4.1).

Lemma 4.3

Let \(\mu _X^\lambda \) be as in (4.2). Then the diffraction measure \(\widehat{\gamma _X^\lambda }=\tau _\lambda (\widehat{\gamma _{X}})\), where \(\tau _\lambda \) is the action of the translation by \(\lambda \); in particular, under (H.2) \(\widehat{\gamma _X^\lambda }\) is singular with respect to Lebesgue measure.

We just sketch the proof, since the methods are standard.

Proof

Note that \(f_\lambda (x)=e^{-2\pi i \langle x,\lambda \rangle }\) is bounded and \(\mu _X\) is translation-bounded, therefore the measure \(\mu _X^\lambda =f_\lambda \mu _X\) is translation-bounded as well, thus an autocorrelation measure \(\gamma _X^\lambda \) exists (cf. [2, Proposition 9.1]). Now definition (1.1) can be applied for \(\mu =\mu _X^\lambda \), and by a change of variable in the formula (1.1) for \(\widehat{\gamma _\mu }\), the first claim follows. Finally, if \(\widehat{\gamma _X}\) is singular with respect to the Lebesgue measure, the second claim follows directly since \(\widehat{\gamma _{X}^{\lambda }}\) is just a translation of \(\widehat{\gamma _X}\) by the vector \(\lambda \in {\mathbb {R}}^d\). \(\square \)

5.1 Independent Random Perturbations and Proof of Theorem A

Proposition 4.4

Let X and \((\xi _p)_{p\in X}\) be as in Theorem A and let \(\mu _\xi ^\lambda \) be as in (4.3). Then almost surely, for all \(\lambda \in {\mathbb {R}}^d\) there holds

$$\begin{aligned} \quad \gamma _\xi ^\lambda = \textsf{Var}\left[ e^{-2\pi i \langle \xi , \lambda \rangle }\right] \textsf{dens}(X)\delta _0. \end{aligned}$$
(4.4)

Proof

Observe that \(\gamma _\xi ^\lambda \) is defined in duality with \(C_c({\mathbb {R}}^d)\) only, as follows:

$$\begin{aligned} \forall f\in C_c({\mathbb {R}}^d)\quad \int f(x)\ d\gamma _\xi ^\lambda (x)= \lim _{R\rightarrow \infty }\frac{1}{Vol(B_R)}\sum _{p,q\in X\cap B_R}f(p-q) \mu _\xi ^\lambda (p)\overline{\mu _\xi ^\lambda (q)}. \end{aligned}$$

We first prove the almost-sure convergence at fixed \(\lambda \). For this we write

$$\begin{aligned} \sum _{p,q\in X\cap B_R}f(p-q) \mu _\xi ^\lambda (p)\overline{\mu _\xi ^\lambda (q)}=f(0)\sum _{p\in X\cap B_R} |\mu _\xi ^\lambda (p)|^2 + \sum _{\begin{array}{c} p,q\in X\cap B_R\\ p\ne q \end{array}}f(p-q) \mu _\xi ^\lambda (p)\overline{\mu _\xi ^\lambda (q)}. \end{aligned}$$

Recall that in (H.1) we assumed that \(\textsf{dens}(X):=\lim _{R\rightarrow \infty }\frac{\#(X\cap B_R)}{Vol(B_R)}\) exists. Therefore, to prove (4.4) it is sufficient to prove that almost surely for \(p_0\) in X (whose precise choice do not matter, since the random variables \(\xi _p\) are assumed to be i.i.d.) and for every compact set \(K\subset {\mathbb {R}}^d\):

$$\begin{aligned} \lim _{R\rightarrow \infty }\frac{1}{\#(X\cap B_R)}\sum _{p\in X\cap B_R}|\mu _\xi ^\lambda (p)|^2= & {} {\mathbb {E}}[|\mu _\xi ^\lambda (p_0)|^2], \end{aligned}$$
(4.5)
$$\begin{aligned} \lim _{R\rightarrow \infty }\frac{1}{Vol(B_R)}\sum _{p\in X\cap B_R}\mu _\xi ^\lambda (p)\sum _{\begin{array}{c} q\in X\cap (K+p)\\ q\ne p \end{array}} \overline{\mu _\xi ^\lambda (q)}= & {} 0. \end{aligned}$$
(4.6)

By applying (4.6) for K a sub-level set for \(\textrm{Re}(f)_-\), \(\textrm{Re}(f)_+\), \(\textrm{Im}(f)_+\), \(\textrm{Im}(f)_-\) and by the Cavalieri principle, it will imply that the sum against \(f(p-q), p\ne q\) as well would tend to zero.

Proof of (4.5). The random variables \(\mu _\xi ^\lambda (p),\mu _\xi ^\lambda (q)\) are deterministic (bounded) functions of \(\xi \), thus are also independent and identically distributed for \(p\ne q\). We find (4.5) by a direct application of the classical strong law of large numbers.

Proof of (4.6). For \(p_0,p_1\in X\) to be distinct fixed points, consider the \({\mathbb {P}}\)-preserving maps \(T_{p,q}:(\Omega , {\mathbb {P}})\rightarrow (\Omega ,{\mathbb {P}})\) induced by the interchange of p with \(p_0\) and of q with \(p_1\), for \(p\ne q\in X\) and

$$\begin{aligned} {{\widetilde{A}}}_R\phi (\omega ):=\frac{1}{Vol(B_R)}\sum _{\begin{array}{c} p\in X\cap B_R\\ q\in X\cap (K+p)\setminus \{p\} \end{array}} \phi (T_{p,q}(\omega )). \end{aligned}$$
(4.7)

Our strategy of proof is to prove convergence of the \(A_R\phi \) as \(R\rightarrow \infty \), after which we can take \(\phi (\omega ):=\mu _\xi ^\lambda (p_0)\overline{\mu _\xi ^\lambda (p_1)}\) to conclude the first equality in (4.6).

Now, we partition the averages from (4.7) over a collection of periodic subsets covering \({\mathbb {R}}^d\), as follows. For each positive integer N, for r being the separation constant of X we set

$$\begin{aligned} C:=\left[ -\frac{r}{2\sqrt{d}},\frac{r}{2\sqrt{d}}\right) ^d, \end{aligned}$$

and for \(i\in \{1\ldots ,N-1\}^d\) we define the set \(A_N^i\) by

$$\begin{aligned} A_N^i:=\bigcup _{k\in \frac{Nr}{\sqrt{d}}{\mathbb {Z}}^d}\left( C+ i\frac{r}{\sqrt{d}} + k\right) . \end{aligned}$$

Note that the sets \(\{A_N^i: i\in \{0,\dots ,N-1\}^d\}\) form a partition of \({\mathbb {R}}^d\), and due to the fact that r is the separation radius of X, for any \(p\ne q\in A_N^i\cap X\) we have \(p-q> (N-1)r/\sqrt{d}\). Assuming N is such that \(\textrm{diam}(K)<(N-1)r/\sqrt{d}\) it follows \(q\notin (K+p)\setminus \{p\}\). Therefore, for such N the random variables

$$\begin{aligned} \left( \sum _{\begin{array}{c} q\in X\cap (K+p)\setminus \{p\} \end{array}} \mu _{\xi }^{\lambda }(p)\overline{\mu _{\xi }^{\lambda }(q)}\right) _{p\in X\cap A_N^i} \end{aligned}$$

are independent. Furthermore, we claim that under the above hypotheses on K, N, if \(\phi \) is bounded and \({\mathbb {E}}[\phi ]=0\) then the following holds: if we define

$$\begin{aligned} A_{R,N}^i\phi (\omega ):=\frac{1}{Vol(B_R)}\sum _{\begin{array}{c} p\in X\cap B_R\cap A_N^i\\ q\in X\cap (K+p)\setminus \{p\} \end{array}} \phi (T_{p,q}(\omega )), \end{aligned}$$

then \(A_{R,N}^i\phi (\omega )\) almost surely converges to 0, i.e

$$\begin{aligned} {\mathbb {P}}\left\{ \omega :\ \lim _{R\rightarrow \infty }A_{R,N}^i\phi (\omega )=0\right\} =1 . \end{aligned}$$
(4.8)

Indeed, we can take \(\phi _j\) to be independent copies of \(\phi \) and write equivalently

$$\begin{aligned} A_{R,N}^i\phi =\frac{n_{R,N}^i}{Vol(B_R)}\cdot \frac{1}{n_{R,N}^i}\sum _{j=1}^{n_{R,N}^i} \phi _j, \qquad n_{R,N}^i:=\#\left\{ (p,q)\in X^2:\ \left. \begin{array}{l}p\in B_R\cap A_N^i\\ q-p\in K\setminus \{0\}\end{array}\right. \right\} . \end{aligned}$$

First note that since X is uniformly discrete and K is bounded, we have that

$$\begin{aligned} \sup _{R>0}\frac{n_{R,N}^i}{Vol(B_R)}<\infty . \end{aligned}$$
(4.9)

Then for each subsequence \(R_k\rightarrow \infty \) such that \(n_{R_k,N}^i/Vol(B_{R_k})\) converges there are two possibilities:

  1. 1.

    The sequence \(n_{R_k,N}^i/ Vol(B_{R_k})\rightarrow 0\), in which case by triangle inequality and using the fact that \(\phi \) is bounded, we find (4.8).

  2. 2.

    We have \(0<\lim _{k\rightarrow \infty }n_{R_k,N}^i/Vol(B_{R_k})\), in particular \(n_{R_k,N}^i\rightarrow \infty \). By boundedness of the sequence \((\phi _j)_{j}\) and by the strong law of large numbers for non-necessarily identically distributed random variables (see for instance [8, Corollary 1]), we get \(\frac{1}{n_{R_k,N}^i}\sum _{j=1}^{n_{R_k,N}^i}\phi _j\rightarrow 0\). Using (4.9), we obtain (4.8) also in this case.

We note that \(\mu _\xi ^\lambda \) is bounded and that

$$\begin{aligned} \mathbb E[\mu _\xi ^\lambda (p_0)\overline{\mu _\xi ^\lambda (p_1)}]=\mathbb E[\mu _\xi ^\lambda (p_0)]\overline{\mathbb E[\mu _\xi ^\lambda (p_1)]}=0\quad \text { for }p_0\ne p_1\in X, \end{aligned}$$

using the fact that the random variables \(\mu _\xi ^\lambda (p_0), \mu _\xi ^\lambda (q_0)\) are i.i.d. and of zero mean. We thus have (4.8) and by summing over i we obtain the desired limit (4.6).

In order to show that the almost sure convergence holds also contemporarily for all \(\lambda \in {\mathbb {R}}^d\), we proceed like in [24]: Firstly, we have that almost-sure convergence can be obtained at for a countable dense subset of \(\lambda \in {\mathbb {R}}^d\). To extend the almost sure convergence to all \(\lambda \), we prove the almost-sure (uniform in \(\lambda \)) control of the \(\lambda \)-gradients

$$\begin{aligned} \limsup _{R>0}\sup _{\lambda \in {\mathbb {R}}^d} \left| \frac{1}{Vol(B_R)}\sum _{p\in X\cap B_R}\sum _{q\in X\cap (K+p)}\nabla \left( \mu _\xi ^\lambda (p)\overline{\mu _\xi ^\lambda (q)}\right) \right| . \end{aligned}$$
(4.10)

For this we just apply triangular inequality, via the rough bounds

$$\begin{aligned} \left| \nabla \mu _\xi ^\lambda (p)\right| \le \left| \xi _p\right| + {\mathbb {E}}[|\xi |], \qquad \left| \mu _\xi ^\lambda (p)\right| \le 2. \end{aligned}$$

We thus find for \(\phi :=\mu _\xi ^\lambda (p_0)\overline{\mu _\xi ^\lambda (q_0)}\):

$$\begin{aligned} \limsup _{R\rightarrow \infty }\sup _{\lambda \in {\mathbb {R}}^d}\left| \nabla {{\widetilde{A}}}_R\phi (\omega )\right| \le \limsup _{R\rightarrow \infty }\frac{2}{Vol(B_R)}\sum _{\begin{array}{c} p\in X\cap B_R\\ q\in X\cap (K+p)\setminus \{p\} \end{array}}(|\xi _p(\omega )|+|\xi _q(\omega )|+2{\mathbb {E}}[|\xi |]). \end{aligned}$$

By partitioning K if it is necessary, we can consider K of small diameter (which does not change the conclusion of (4.6)), and then \(X\cap (K+p)\) has cardinality at most 1 for all p. Thus

$$\begin{aligned} \sum _{\begin{array}{c} p\in X\cap B_R\cap A_N^i\\ q\in X\cap (K+p)\setminus \{p\} \end{array}}(|\xi _p(\omega )|+|\xi _q(\omega )|+2{\mathbb {E}}[|\xi |])\le 2\sum _{p\in X\cap (B_R+K)}(|\xi _p|+{\mathbb {E}}[|\xi |]), \end{aligned}$$
(4.11)

and now we can label points \(\{p_n\}_{n\in {\mathbb {N}}}= X\cap A_N^i\) so that \(|p_n|\) is increasing, and apply the classical strong law of large numbers to the random variables \(|\xi _{p_n}|\) to show that the right hand side in (4.11) is almost surely bounded by \(C_X{\mathbb {E}}[|\xi |]Vol(B_R)\). This gives that almost surely (4.10) is bounded by \(C_X{\mathbb {E}}[|\xi |]\) in which \(C_X\) is a packing constant, \(C_X\le \frac{C_d}{r^d}\) in which \(C_d\) depends only on the dimension and \(r>0\) is the uniform separation radius of X.

Note that, if we fix a countable dense \(\Lambda \subset {\mathbb {R}}^d\), almost surely (4.5) and (4.6) hold for every \(\lambda \in \Lambda \). Thus, to get the almost sure convergence in (4.6) contemporarily also for all \(\lambda \in {\mathbb {R}}^d\setminus \Lambda \), observe that if \((\lambda _n)_{n\in {\mathbb {N}}}\subset \Lambda \) is a sequence such that \(\lambda _n\longrightarrow \lambda \) , then for \(\phi ^{\lambda }=\mu _{\xi }^{\lambda }(p_0)\overline{\mu _{\xi }^{\lambda }(q_0)}\) there holds

$$\begin{aligned} \begin{aligned} \left| {\widetilde{A}}_R\phi ^{\lambda }\omega \right|&\le \left| {\widetilde{A}}_R\phi ^{\lambda }\omega -{\widetilde{A}}_R\phi ^{\lambda _n}\omega \right| +\left| {\widetilde{A}}_R\phi ^{\lambda _n}\omega \right| \\&\le M_K|\lambda -\lambda _n|+\left| {\widetilde{A}}_R\phi ^{\lambda _n}\omega \right| , \end{aligned} \end{aligned}$$

where \(M_K\) is the value of (4.10). By taking the limsup as \(R\rightarrow \infty \) and then \(n\rightarrow \infty \) in the last inequality, we obtain that on the same event on which convergence holds for the \(\lambda _n\) we also have \({\widetilde{A}}_R\phi ^{\lambda }\longrightarrow 0\). Thus this convergence holds almost surely for every \(\lambda \in {\mathbb {R}}^d\). Finally, an analogous argument together an application of the \(\lambda \)-continuity of \(\mu _{\xi }^{\lambda }\) and the dominated convergence theorem, we get the convergence in (4.5) almost surely for all \(\lambda \). This concludes the proof of Proposition 4.4. \(\square \)

Proof of Proposition 4.2

From Theorem A.4, Lemma 4.3 and Proposition 4.4 (resp. Propositions 4.7 and 4.16 below in the case of Theorems B and C), we have that for every positive test function \(f\in {\mathcal {S}}({\mathbb {R}}^d)\)

$$\begin{aligned} \lim _{R\rightarrow \infty }\dfrac{1}{Vol(B_R)}\left| \displaystyle \sum _{p\in X\cap B_R}\mu _{\xi }^{\lambda }(p)\overline{\mu _X^{\lambda }(p)}{\widehat{f}}(0)+\sum _{\begin{array}{c} p,q\in X\cap B_R\\ q\ne p \end{array}}\mu _{\xi }^{\lambda }(p)\overline{\mu _X^{\lambda }(q)}{\widehat{f}}(p-q)\right| =0.\nonumber \\ \end{aligned}$$
(4.12)

In particular, we can choose \(f=f_\epsilon \in {\mathcal {S}}({\mathbb {R}}^d)\) such that \(1<\widehat{f_{\varepsilon }}(0)\) (in effect, what we use below is that \(0<\widehat{f_\varepsilon }(0)\) and \(\varepsilon /\widehat{f_\varepsilon }(0)\rightarrow 0\), for \(\epsilon \rightarrow 0\)) and

$$\begin{aligned} \forall p\in X\qquad \sum _{q\in X\setminus \{p\}}|{\widehat{f}}_\epsilon (p-q)|<\epsilon . \end{aligned}$$

This is possible by the uniform discreteness of X, by taking \({{\hat{f}}}_\epsilon \) radially decreasing and concentrated on B(0, r/2) where r is the separation constant of X in the sense that \(\int _{{\mathbb {R}}^d\setminus B(0,r/2)} |{{\hat{f}}}_\epsilon |< \frac{r^d\epsilon }{C}\). For example, this holds for \({{\hat{f}}}_\epsilon \) a Gaussian centered at 0 with small enough variance. For such \(f_\epsilon \) we can estimate \(\sum _{q\in X\setminus \{p\}}|{{\hat{f}}}_\epsilon (p-q)|\) by the abovementioned integral, using the separation of X and the fact that \({{\hat{f}}}_\epsilon \) is radially decreasing.

Now fix \(R_0\) such that the quantity in (4.12), is smaller than \(\epsilon \) and that \(\frac{\#(X\cap B_R)}{Vol(B_R)}<\textrm{dens}(X)+1\) for all \(R\ge R_0\). By triangle inequality applied to (4.12), for such R we get

$$\begin{aligned} \dfrac{\widehat{f_{\varepsilon }(0)}}{Vol(B_R)}\left| \displaystyle \sum _{p\in X\cap B_R}\mu _{\xi }^{\lambda }(p)\overline{\mu _X^{\lambda }(p)}\right| \le \varepsilon +\dfrac{2}{Vol(B_R)}\sum _{\begin{array}{c} p,q\in X\cap B_R\\ q\ne p \end{array}}|{\widehat{f}}_\epsilon (p-q)|\le (2\ \textrm{dens}(X)+3)\ \epsilon , \end{aligned}$$

which follows by noting that \(\sup _{p\ne q\in X}|\mu _\xi ^\lambda (p)\overline{\mu _X^\lambda (q)}|\le 2\). By dividing by \(\widehat{f_{\varepsilon }(0)}\), we then take \(\epsilon \rightarrow 0\) in the above inequality, and this completes the proof of Proposition 4.2 in the case of i.i.d random perturbations.

5.2 Non-independent Random Perturbations and Proof of Theorem B

For extending Proposition 4.4 to the case of non-independent identically distributed \(\xi _p,p\in X\), we need a mixing hypothesis. Several notions of mixing are available (see the survey [7]); we perform our proofs based on the following ad-hoc strong mixing condition. For \(p\in X\) and \(K\subset {\mathbb {R}}^d\) compact, define the sub-\(\sigma \)-algebras

$$\begin{aligned} {\mathcal {F}}_p:=\sigma (\xi _p),\quad {\mathcal {F}}_{p,K}:=\sigma (\xi _q:\ q \in (K+p)\cap X\setminus \{p\}). \end{aligned}$$

we say that the random process \((\xi _p)_{p\in X}\) is compactly-mixing if for any fixed compact \(K\subset {\mathbb {R}}^d\) there holds

$$\begin{aligned} \lim _{R\rightarrow \infty }\frac{1}{Vol(B_R)}\sum _{p\in X\cap B_R} \sup \left| {\mathbb {E}}[fg]- {\mathbb {E}}[f]{\mathbb {E}}[g]\right| =0. \end{aligned}$$
(4.13)

where for each \(p\in X\) the supremum is taken over the set of all functions \(f\in L^{\infty }({\mathcal {F}}_p,{\mathbb {P}})\), \(g\in L^{\infty }({\mathcal {F}}_{p,K},{\mathbb {P}})\) such that \(||f||_{L^{\infty }}\le 1\), \(||g||_{L^{\infty }}\le 1\). Observe that (4.13) is equivalent to

$$\begin{aligned} \lim _{R\rightarrow \infty }\frac{1}{Vol(B_R)}\sum _{p\in X\cap B_R} \sup \left| {\mathbb {P}}[A\cap B]- \mathbb {P}[A]{\mathbb {P}}[B]\right| =0, \end{aligned}$$

where for each \(p\in X\), the supremum is taken over the events \(A\in {\mathcal {F}}_p\) and \(B\in {\mathcal {F}}_{p,K}\).

We first prove the following:

Lemma 4.5

For \(\mu _{\xi }^{\lambda }\) as before and for \(\lambda \in {\mathbb {R}}^d\) fixed, we have that almost surely

$$\begin{aligned} \frac{1}{Vol(B_R)}\sum _{p\in X\cap B_R}\sum _{\begin{array}{c} q\in X\cap (K+p)\\ q\ne p \end{array}}\left[ \mu _{\xi }^{\lambda }(p)\overline{\mu _{\xi }^{\lambda }(q)}-{\mathbb {E}}[\mu _{\xi }^{\lambda }(p)\overline{\mu _{\xi }^{\lambda }(q)}]\right] \longrightarrow 0. \end{aligned}$$
(4.14)

Proof

To prove this, we apply Proposition A.1 to the real and imaginary parts of the variables \(\mu _\xi ^\lambda (p)\overline{\mu _\xi ^\lambda (q)}\) corresponding to elements \(p,q\in X\) such that \(p-q\in K\). Firstly, by (H.1) we may replace the factor \(1/Vol(B_R)\) by \(1/\#(X\cap B_R)\) in (4.14). Note that the sets of index pairs figuring in the double sums in (4.14) form a sequence increasing in R, thus we can find a labelling of \(X_i:=\mu _\xi ^\lambda (p)\overline{\mu _\xi ^\lambda (q)}\) so that the convergence in (4.14) follows from the convergence of a subsequence of the \(\frac{1}{n}\sum _{n=1}^n(X_k -{\mathbb {E}}[X_k])\). We apply Proposition A.1 to obtain convergence of the full sequence. We verify the hypotheses of Proposition A.1 next, for this case:

  • The bounds \(C_k\) and the control of part 1 of the hypotheses of Proposition A.1 follow because \(|\mu _\xi ^\lambda (p)|\le 2\) almost surely.

  • For the summability of variances along geometric series of hypothesis 2 of Proposition A.1, it suffices to prove \(\frac{\textsf{Var}(S_{k_n})}{k_n}\le C\), which follows from the boundedness of \(\textsf{Var}(\mu _{\xi }^{\lambda }(p)\overline{\mu _{\xi }^{\lambda }(q)})\) and \(\textsf{Cov}(\mu _{\xi }^{\lambda }(p),\mu _{\xi }^{\lambda }(q))\). In fact, for \(X_i=\mu _{\xi }^{\lambda }(p)\overline{\mu _{\xi }^{\lambda }(q)}\) as before, we get:

    $$\begin{aligned} \textsf{Var}(S_{k_n})=\sum _{i=1}^{k_n}\textsf{Var}(X_i)+2\sum _{\begin{array}{c} i,j=1\\ i\ne j \end{array}}^{k_n}\textsf{Cov}(X_i,X_j), \end{aligned}$$

    and by the definition of \(\mu _{\xi }^\lambda (p)\) we have

    $$\begin{aligned} \begin{aligned} \textsf{Var}(X_i)&\le {\mathbb {E}}\left[ \left| \mu _{\xi }^{\lambda }(p)\overline{\mu _{\xi }^{\lambda }(q)}\right| ^2\right] \le 16\\ |\textsf{Cov}(X_i,X_j)|&=\left| {\mathbb {E}}\left( \mu _{\xi }^{\lambda }(p)\overline{\mu _{\xi }^{\lambda }(q)}\overline{\mu _{\xi }^{\lambda }(p')}\mu _{\xi }^{\lambda }(q')\right) -{\mathbb {E}}\left[ \mu _{\xi }^{\lambda }(p)\overline{\mu _{\xi }^{\lambda }(q)}\right] {\mathbb {E}}\left[ \overline{\mu _{\xi }^{\lambda }(p')}\mu _{\xi }^{\lambda }(q')\right] \right| \\&\le 32. \end{aligned} \end{aligned}$$

Therefore, the limit (4.14) follows as a consequence of Proposition A.1 as claimed. \(\square \)

Lemma 4.6

If the random process \((\xi _p)_{p\in X}\) is compactly-mixing, then there holds

$$\begin{aligned} \sum _{p\in X\cap B_R}\sum _{\begin{array}{c} q\in X\cap (K+p)\\ q\ne p \end{array}}{\mathbb {E}}[\mu _{\xi }^{\lambda }(p)\overline{\mu _{\xi }^{\lambda }(q)}]=o(R^d). \end{aligned}$$
(4.15)

Proof

Indeed, this a direct consequence of the definition (4.13), observing that \(\mu _\xi ^\lambda (p)\) is almost surely bounded for all \(\lambda \). \(\square \)

Now (4.14) and (4.15) give the following:

Proposition 4.7

Assume that the \((\xi _p)_{p\in X}\) are identically distributed and compactly-mixing, and let \(\mu _\xi ^\lambda \) be as in (4.3). Then almost surely, for all \(\lambda \in {\mathbb {R}}^d\) there holds

$$\begin{aligned} \gamma _{\xi }^{\lambda }=\textsf{Var}\left[ e^{-2\pi i\langle \xi ,\lambda \rangle }\right] \textsf{dens}(X)\delta _0. \end{aligned}$$

To obtain Proposition 4.2 under the above hypotheses, we now note that the upper bound of (4.10) can also be extended under our assumed bound on the covariances of the \(\xi _p\). Indeed all the proof up to (4.11) can be repeated verbatim, and for the bound of the last term in (4.11), we may now apply Proposition A.2 to the \(|\xi _p|\) after to enumerate the points in \(X\cap B_R\) and by using (H.1), which gives a law of strong numbers under our assumed condition (2.2) on the covariances of the \(|\xi _p|\). This concludes the proof of Theorem B.

Example 4.8

Let \((R_k)_{k\ge 1}\) be a sequence of positive real numbers such that \(R_k,R_k/R_{k-1}\longrightarrow \infty \), and denote \(\Omega _k:=B_{R_k}\setminus B_{R_{k-1}}\) for \(k\ge 2\). Consider a process of identically distributed random vectors \((\xi _p)_{p\in X}\) such that the following hypotheses on correlations hold.

  1. 1.

    For every \(p\in X\cap B_{R_1}\), the random vector \(\xi _p\) has no correlation with the \(\xi _{p'}, p'\in X\setminus \{p\}\).

  2. 2.

    For every even \(k>1\), the \(\xi _p, p\in X\cap \Omega _k\) are not correlated to the remaining \(\xi _{p'}, p'\in X\setminus \Omega _k\).

  3. 3.

    For every odd \(k>1\) and for for \(p\ne p'\in X\cap \Omega _k\), \(\xi _p\) and \(\xi _{p'}\) are uncorrelated, and for every \(p\in X\cap \Omega _k\), there exists at least one \(q\in X\cap \Omega _{k-1}\) such that \(\xi _p \) has nontrivial correlation to the value of \(\xi _q\).

It is not hard to check that this procedure produces a random process satisfying (4.13), since correlations at distance closer than \(\textrm{diam}(K)\) do not occur.

As a further remark, which allows to understand the role of the hypothesis that the \(\xi _p\) are compactly-mixing, which is used here in order to obtain (4.15). The following result makes it explicit, that the outcome of this is to obtain a version of (4.4), but in expectation rather than almost-surely. As a consequence, we could also require as a hypothesis the control stated (4.16) below, rather than the compact-mixing hypothesis.

Lemma 4.9

If \(\xi _p\) are identically distributed and if X verifies (H1) then the fact that (4.15) holds for all \(\lambda \in {\mathbb {R}}^d\) is equivalent to the following:

$$\begin{aligned} \forall \lambda \in {\mathbb {R}}^d,\qquad \mathbb E(\gamma _\xi ^\lambda )=\textsf{Var}\left[ e^{-2\pi i\langle \xi ,\lambda \rangle }\right] \textrm{dens}(X)\delta _0. \end{aligned}$$
(4.16)

Proof

Indeed, we have

$$\begin{aligned} \lim _{R\rightarrow \infty } \frac{1}{Vol(B_R)}\sum _{p\in X\cap B_R} \mathbb E\left[ |\mu _\xi ^\lambda (p)|^2\right] =\textrm{dens}(X)\textsf{ Var}(\mu _\xi ^\lambda ), \end{aligned}$$

thus (4.15) is equivalent to

$$\begin{aligned} \sum _{p\in X\cap B_R}\sum _{q\in X\cap (K+p)}{\mathbb {E}}\left[ \mu _{\xi }^{\lambda }(p)\overline{\mu _{\xi }^{\lambda }(q)}\right] =\textrm{dens}(X)\ Vol(B_R)\textsf{Var}(\mu _\xi ^\lambda )+o(R^d). \end{aligned}$$
(4.17)

The fact that (4.17) is valid for all compact K is equivalent to the convergence of the corresponding \((Vol(B_R))^{-1}\)-rescaled sum tested against all test functions f, which is the weak formulation of (4.16). \(\square \)

5.3 Extension of Theorems A, B for X Being the Asymptotically Affine Deformation of a Lattice

In this section we extend Theorems A, B by weakening the independence requirements on the random perturbations \(\xi _p\). To achieve this we limit ourselves to a smaller class of sets X.

A useful tool to replace the independence of the \(\xi _p\), \(p\in X\), is the notion of ergodicity of the process \((\xi _p)_{p\in X}\), labelled by the set X. To make sense of this notion, commonly X is required to have more structure, e.g a transitive group action on X is required, and classically X itself is assumed to be a lattice. Therefore, firstly we study the case of X being a lattice: in this case we recover the main results in [24]. Secondly, we consider the case when X is almost-affine, meaning that though X lacks a group structure (the sum of elements of X is not generally in X), it is a small perturbation of a subgroup of \({\mathbb {R}}^d\). In this case the results still depend strongly on a group structure. For a discussion of further weakenings on the structure of X, see Sect. 5.1.

5.3.1 Lattice Case

As was pointed out by Yakir in [24], in the case of X being a lattice, the condition that the process \((\xi _p)_{p\in X}\) is ergodic is sufficient for (5.1) to converge as \(R\rightarrow \infty \).

Lemma 4.10

(Lattice case) If X is a lattice and \((\xi _p)_{p\in X}\) is stationary and ergodicwith respect to lattice shifts, then almost surely

$$\begin{aligned} \gamma _{\xi }^{\lambda }=\sum _{k\in X}\mathbb E\left[ \mu _\xi ^\lambda (0)\overline{\mu _{\xi }^{\lambda }(k)}\right] \textrm{dens}(X)\delta _k. \end{aligned}$$
(4.18)

Proof

By stationarity and using Birkhoff’s theorem for the natural shifts on X, we ensure the existence of the limit defining \(\gamma _\xi ^\lambda (k)\) as in (5.1). Ergodicity of the process \((\xi _p)_{p\in X}\) yields that the above limit is almost surely constant and thus almost surely converges to \({\mathbb {E}}[\mu _{\xi }^{\lambda }(0)\overline{\mu _{\xi }^{\lambda }(k)}]\). \(\square \)

In general a measure like \(\gamma _\xi ^\lambda \) as given in (4.18) may have Fourier transform \(\widehat{\gamma _\xi ^\lambda }\) with nonzero atomic part (i.e. the initial \(\sum _{p\in X}\mu _\xi ^\lambda (p)\delta _p\) may have pure-point spectrum). If however we strengthen the ergodicity condition on the map \((\xi _p)_{p\in X}\) to a weak mixing condition, this is precluded. We require the following weak mixing condition. Let \(\tau _k: X\rightarrow X\) be the translation by \(k\in X\). Associated to each \(\tau _k\) we define a measurable \(\mathbb P\)-preserving transformation \(T_k\), which is induced by interchanging the X-coordinates \((\xi _p)_{p\in X}\) according to \(\tau _k\). The process \((\xi _p)_{p\in X}\) is weakly mixing if for all \(f,g\in L^2({\mathbb {P}})\) there holds

$$\begin{aligned} \lim _{R\rightarrow \infty }\frac{1}{Vol(B_R)}\sum _{k\in X\cap B_R}\left| {\mathbb {E}}[f\circ T_k\ {{\overline{g}}}] - \mathbb E[f]\overline{{\mathbb {E}}[g]}\right| =0. \end{aligned}$$
(4.19)

Note that the compactly-mixing condition introduced earlier in (4.13) is not directly comparable to (4.19), and we do not pursue an in-depth comparison in this work.

We have the following

Lemma 4.11

(Basic lattice case) If X is a lattice and \((\xi _p)_{p\in X}\) is stationary and weakly mixing then almost surely for every \(\lambda \in {\mathbb {R}}^d\) the random diffraction measure \(\widehat{\gamma _\xi ^\lambda }\) has no atoms.

Proof

Recall Strungaru’s criterion [22, Proposition 4.1] (here formulated in the setting of formula (4.18) which we know to hold for \(\gamma _\xi ^\lambda \)):

$$\begin{aligned} \lim _{R\rightarrow \infty }\frac{1}{Vol(B_R)}\sum _{k\in X\cap B_R}\left| \mathbb E[\mu _\xi ^\lambda (0)\overline{\mu _\xi ^\lambda (k)}]\right| =0\quad \Leftrightarrow \quad \widehat{\gamma _\xi ^\lambda } \text { has no atoms}. \end{aligned}$$
(4.20)

We note that (4.20) follows from (4.19), if we apply it for \(f(\omega )=g(\omega )=e^{2\pi i\langle \lambda , \xi _0(\omega )\rangle }\), since by stationarity \(\xi _k=\xi _0\circ T_k\) and

$$\begin{aligned} \mathbb E\left[ \mu _\xi ^\lambda (0)\overline{\mu _\xi ^\lambda (k)}\right] =\mathbb E\left[ e^{-2\pi i\langle \xi _0-\xi _k, \lambda \rangle }\right] - {\mathbb {E}}\left[ e^{-2\pi i\langle \xi _0,\lambda \rangle }\right] \overline{{\mathbb {E}}\left[ e^{-2\pi i\langle \xi _k,\lambda \rangle }\right] }. \end{aligned}$$

\(\square \)

5.3.2 Asymptotically Affine Perturbations of a Lattice

Now assume that \(X=\psi (L)\) where L is a lattice, and that \(\xi _p\) are identically distributed and satisfy \(\mathbb E(|\xi _p|^{d+\epsilon })<\infty \) as before. Denote \(\mu '(x):=\mu _\xi ^\lambda (\psi (x))\) for \(x\in L\), and assume that \(\mu '(x)\) are stationary and ergodic under the natural shifts of \(x\in L\). We have the following asymptotic equalities, valid in the limit as \(R\rightarrow \infty \), with error estimates as in Lemma 4.1:

$$\begin{aligned}{} & {} {\frac{1}{Vol(B_R)}\sum _{p,q\in X\cap B_R} g(p-q)\mu _\xi ^\lambda (p)\overline{\mu _\xi ^\lambda (q)}}\nonumber \\{} & {} \quad \simeq \frac{1}{Vol(B_R)}\sum _{x,y\in L\cap B_R}g(\psi (x)-\psi (y))\mu '(x)\overline{\mu '(y)}\nonumber \\{} & {} \quad \simeq \sum _{k\in L\cap \textrm{supp}(g)}\frac{1}{Vol(B_R)}\sum _{x\in L\cap B_R} g(\psi (x)-\psi (x+k)) \mu '(x)\overline{\mu '(x+k)} \end{aligned}$$
(4.21)

We find it natural based on (4.21) to introduce the following notion. We say that \(\psi \) has asymptotically affine if one of the following equivalent conditions holds:

  1. 1.

    There exists \(F:L\rightarrow {\mathbb {R}}\) such that for any continuous compactly supported g and for any \(k\in L\)

    $$\begin{aligned} \lim _{R\rightarrow \infty } \frac{1}{Vol(B_R)}\sum _{x\in L\cap B_R}g(\psi (x)-\psi (x+k))= g(F(k)). \end{aligned}$$
    (4.22)
  2. 2.

    For all \(k\in L\) there holds

    $$\begin{aligned} \lim _{\begin{array}{c} x\in L\\ |x|\rightarrow \infty \end{array}}(\psi (x)-\psi (x+k))=F(k). \end{aligned}$$
    (4.23)
  3. 3.

    The limit (4.23) holds for a linear function F.

To see that \(1\Rightarrow 2\) above, it suffices to test (4.22) over a sequence of \(g_n\) that approximate a Dirac mass at F(k). Then \(2\Rightarrow 3\) follows by observing that if the limit in (4.23) exists for \(k\in \{k_1,k_2\}\) then

$$\begin{aligned}F(k_1)+F(k_2)= & {} \lim _{\begin{array}{c} x\in L\\ |x|\rightarrow \infty \end{array}}(\psi (x)-\psi (x+k_1))+\lim _{\begin{array}{c} x\in L\\ |x|\rightarrow \infty \end{array}}(\psi (x+k_1)-\psi (x+k_1+k_2))\\= & {} F(k_1+k_2), \end{aligned}$$

and finally, \(3\Rightarrow 1\) is a direct verification.

Now if we assume that \(\psi \) satisfies (4.22) and we insert this limit into (4.21), using the fact that \(\mu '(x)\) are bounded and that \(\textrm{supp}(g)\cap L\) is finite, we directly find

$$\begin{aligned} \text {(4.21)}\simeq & {} \sum _{k\in L\cap \textrm{supp}(g)}\frac{1}{Vol(B_R)}\sum _{x\in L\cap B_R} g(F(k)) \mu '(x)\overline{\mu '(x+k)}\nonumber \\\simeq & {} \sum _{k\in L\cap \textrm{supp}(g)}\mathbb E[\mu _\xi ^\lambda (0)\overline{\mu _\xi ^\lambda (k)}]\textrm{dens}(L)g(F(k)), \end{aligned}$$
(4.24)

where in the second step we applied Lemma 4.10. We thus have from (4.21) directly the following result, which extends Lemmas 4.10 and 4.11:

Proposition 4.12

(asymptotically affine perturbations of a lattice) Assume that \(X=\psi (L)\) where L is a lattice and \(\psi \) is asymptotically affine with limit linear part F. Assume that the random process \(\xi _{\psi (x)}\), \(x\in L\) is stationary and ergodic under shifts of L, and that the \(\xi _{\psi (x)}\) are identically distributed. Then the limit defining \(\gamma _\xi ^\lambda \) as in (5.1) exists and we have

$$\begin{aligned} \gamma _\xi ^\lambda =\sum _{k\in L}\mathbb E\left[ \mu _\xi ^\lambda (0)\overline{\mu _\xi ^\lambda (k)}\right] \textrm{dens}(L)\delta _{F(k)}. \end{aligned}$$
(4.25)

Furthermore, if \(\xi _{\psi (x)}, x\in L\) is weakly-mixing under shifts of L, then \(\widehat{\gamma _\xi ^\lambda }\) has no atoms.

The above result directly allows to obtain Proposition 4.2, which as before, together with Lemma 4.1 gives the following result:

Theorem D

For \(X=\psi (L)\subset {\mathbb {R}}^d\) as in Proposition 4.12, assume that the random process \(\xi _{\psi (x)}\), \(x\in L\) is stationary and weakly mixing under shifts of L, that the \(\xi _p\)’s are identically distributed and that there exists a positive number \(\varepsilon \) such that \({\mathbb {E}}(|\xi |^{d+\varepsilon })<\infty \). Then, almost surely for every \(\lambda \in {\mathbb {R}}^d\), in the sense of vague convergence of measures, the same limit (2.1) as in Theorem A holds.

5.4 Stationary Random Perturbations and Proof of Theorem C

5.4.1 Stationary Random Measures

The proof of Theorem C combines the recovery problem setting in which \(X_\xi =\{p+\xi (p):\ p\in X\}\) as before, with the study of [11] who established a diffraction theory for stationary point processes. The setting of point processes is not sufficient for our setting, therefore we first generalize Gouéré’s work to locally finite random measures.

We start with a series of definitions and properties that are classical for point processes, see e.g. [18]. The case of point processes is the special case in which the random measures \(\mu \) below are the empirical measures of random sets \(X\in {\mathbb {R}}^d\), i.e. \(\mu =\sum _{x\in X}\delta _x\). Proofs valid for this special case directly generalize to our setting and are well-known, therefore we only sketch the proofs.

For any \(f\in C_c({\mathbb {R}}^d)\) define the functions \(N_f:{\mathcal {M}}^{\infty }\rightarrow {\mathbb {C}}\) by \(N_f(\mu ):=\mu (f)=\langle \mu ,f\rangle \). Let \({\mathcal {A}}\) be the \(\sigma \)-algebra on \({\mathcal {M}}^{\infty }\) generated by the \(N_f\)’s. A random measure is a measurable function \(\mu :\Omega \rightarrow {\mathcal {M}}^{\infty }\) defined on a probability space \((\Omega ,{\mathcal {F}},{\mathbb {P}})\) with values in \(({\mathcal {M}}^{\infty },{\mathcal {A}})\). We say that a random measure is integrable if the functions \(N_f\) are integrable. The random measure is called stationary if for every \(x_1,\ldots ,x_k\in {\mathbb {R}}^d\) and every \(F_1,\ldots ,F_k\in {\mathcal {A}}\),

$$\begin{aligned}{} & {} {\mathbb {P}}\{\omega \in \Omega :\ \mu *\delta _{x_1+t}\in F_1,\ldots ,\mu *\delta _{x_k+t}\in F_k\}\\{} & {} \quad ={\mathbb {P}}\{\omega \in \Omega :\ \mu *\delta _{x_1}\in F_1,\ldots ,\mu *\delta _{x_k}\in F_k\} \end{aligned}$$

holds for every \(t\in {\mathbb {R}}^d\).

Let \(\mu \) be a stationary and integrable random measure. The Palm measure of \(\mu \) is the measure \({\widetilde{P}}\) on \(({\mathcal {M}}^{\infty },{\mathcal {A}})\) defined by

$$\begin{aligned} {\widetilde{P}}(F):=\frac{1}{Vol(B_1)}\cdot {\mathbb {E}}\left( \sum _{x\in \textsf{supp}(\mu )\cap B_1}\mathbbm {1}_F(\mu *\delta _{-x})\right) , F\in {\mathcal {A}}. \end{aligned}$$

The intensity \(I({\widetilde{P}})\) of the Palm measure \({\widetilde{P}}\) is the functional on \(C_c({\mathbb {R}}^d)\) defined by

$$\begin{aligned} I({\widetilde{P}})(f):=\int _{{\mathcal {M}}^{\infty }}\mu (f)d{\widetilde{P}}(\mu ). \end{aligned}$$

Lemma 4.13

(Cf. [18, Prop. 2.24]) Let \(\mu \) be a stationary and integrable random measure. The intensity \(I({\widetilde{P}})\) of its Palm measure is a translation-bounded (and therefore tempered) measure.

Let \(\psi \in C_c({\mathbb {R}}^{d})\) be a non-negative function such that \(\textsf{supp}(\psi )=B_M\), \(\int \psi (x)dx=1\), and which will be kept fixed from now on. Let \({\mathcal {M}}^{\infty }_{X}\) be the set of measures \(\mu \in {\mathcal {M}}^{\infty }\) for which there is \(t\in {\mathbb {R}}^d\) such that \(\textsf{supp}(\mu )= X+t\). For every \(f\in C_c({\mathbb {R}}^d)\), define the function \(H_f:{\mathcal {M}}^{\infty }_{X}\rightarrow {\mathbb {C}}\) by

$$\begin{aligned} H_f(\nu ):=\sum _{x,y\in \textsf{supp}(\nu )}\psi (x)\nu (x)\overline{\nu (-y)}f(x-y). \end{aligned}$$

The next lemma expresses the autocorrelation of a deterministic measure \(\nu \in M_X^{\infty }\) as a limit of orbit averages defined on \(M_X^{\infty }\). This result extends [11, Lem. 2.13] to general locally finite atomic measures.

Lemma 4.14

Let \(\gamma \) be a locally finite measure on \({\mathbb {R}}^d\). Then \(\gamma \) is the autocorrelation \(\gamma _\nu \) of a (deterministic) measure \(\nu \in {\mathcal {M}}^{\infty }_{X}\) if and only if for all \(f\in C_c({\mathbb {R}}^d)\),

$$\begin{aligned} \lim _{R\rightarrow \infty }\frac{1}{Vol(B_R)}\int _{B_R}H_f(\nu *\delta _{-t})dt=\gamma (f). \end{aligned}$$

Proof

As \(\gamma \) is translation bounded, it is sufficient to show that for all \(f\in C_c({\mathbb {R}}^d)\) (see Lemma 1.2 in [20])

$$\begin{aligned} \frac{1}{Vol(B_R)}\left| \int _{B_R}H_f(\nu *\delta _{-x})dt-\sum _{\begin{array}{c} x\in \textsf{supp}(\nu )\cap B_R\\ y\in \textsf{supp}(\nu ) \end{array}}\nu (x)\overline{\nu (-y)}f(x-y)\right| \longrightarrow 0. \end{aligned}$$
(4.26)

A simple computation shows that

$$\begin{aligned} {\mathcal {H}}_f(\nu *\delta _{-t})=\sum _{x,y\in \textsf{supp}(\nu )}\psi (x-t)\nu (x)\overline{\nu (-y)}f(x-y). \end{aligned}$$

On the other hand, for all \(R\ge M\) we have the following properties, since \(\textrm{supp}\psi \subset B_M\)

  • If \(x\not \in B_{R+M}\), then \(\int _{B_R}\psi (x-t)dt=0\), and

  • if \(x\in B_{R-M}\), then \(\int _{B_R}\psi (x-t)dt=1\).

Hence, the left-hand side of (4.26) is equal to

$$\begin{aligned} \begin{aligned} \frac{1}{Vol(B_R)}\left| \sum _{\begin{array}{c} x\in \textsf{supp}(\nu )\cap B_{R+M}\setminus B_{R-M}\\ y\in \textsf{supp}(\nu ) \end{array}}f(x-y)\nu (x)\overline{\nu (-y)}\int _{B_{R}}\psi (x-t)dt\right. \\ -\left. \sum _{\begin{array}{c} x\in \textsf{supp}(\nu )\cap B_R\setminus B_{R-M}\\ y\in \textsf{supp}(\nu ) \end{array}}\nu (x)\overline{\nu (-y)}f(x-y)\right| . \end{aligned} \end{aligned}$$

By triangle inequality, this last expression is bounded by

$$\begin{aligned} \begin{aligned} \frac{||f||_{\infty }C_{\nu }}{Vol(B_R)}\left( \sum _{x\in \textsf{supp}(\nu )\cap B_{R+M}\setminus B_{R-M}}\#(\textsf{supp}(\nu )\cap (\textsf{supp}(f)+x))\right. \\ \left. +\sum _{x\in \textsf{supp}(\nu )\cap B_{R}\setminus B_{R-M}}\#(\textsf{supp}(\nu )\cap (\textsf{supp}(f)+x))\right) , \end{aligned} \end{aligned}$$
(4.27)

where \(|\nu (x)\overline{\nu (-y)}|\le C_{\nu }\) is a positive constant only depending on \(\nu \). Moreover since \(\nu \in {\mathcal {M}}^{\infty }_{X}\), \(f\in C_c({\mathbb {R}}^d)\) and by the discreteness of X, it follows that \(\#(\textsf{supp}(\nu )\cap (\textsf{supp}(f)+x))\) can be bounded by a positive packing constant \(C_f\) only depending on \(\textrm{diam}(\textsf{supp}(f))\) and on the separation constant of X. Thus, by writting \(D_f=C_f||f||_{\infty }\), we have that (4.27) can be bounded from above by

$$\begin{aligned} \frac{D_fC_{\nu }}{Vol(B_R)}(\#(\textsf{supp}(\nu )\cap B_{R+M}\setminus B_{R-M})+\#(\textsf{supp}(\nu )\cap B_R\setminus B_{R-M})). \end{aligned}$$
(4.28)

Note that (H.1) implies that for each \(M>0\) and for each \(p\in {\mathbb {R}}^d\) there holds

$$\begin{aligned} \lim _{R\rightarrow \infty }\frac{\# \left( X\cap B_R(p)\setminus B_{R-M}(p)\right) }{Vol(B_R)}=0. \end{aligned}$$
(4.29)

The proof of (4.29) is standard: we compare \(\#(X\cap B_R(p))\) with the values \(\#(X\cap B_{R\pm |p|})\) and compare the latter with \(\textrm{dens}(X) Vol(B_{R\pm |p|})\) via (H.1); then we do the same with R replaced by \(R-M\). Due to the precise power law growth \(Vol(B_R)=C R^d\) and to (H.1), all the error terms are \(o(R^d)\) or \(O(R^{d-1})\), which implies (4.29).

Since \(\nu \in {\mathcal {M}}^{\infty }_{X}\), from property (4.29) of X we have that (4.28) converges to zero when R goes to infinity. Therefore the limit in (4.26) holds, as desired. \(\square \)

The following result was also proved in [11] for point processes, and we note that it extends to random measures, using Lemma 2.13 therein, and it directly extends via Lemma 4.14.

Proposition 4.15

(Cf. [11, Theorem 4.3]) Let \(\mu \) be a stationary and integrable random measure with values in \({\mathcal {M}}^{\infty }_X\). Then \(\mu \) has almost surely a (random) autocorrelation \(\gamma _{\mu }\) such that

$$\begin{aligned} {\mathbb {E}}(\gamma _{\mu })=I({{\widetilde{P}}} ) \text{ and } {\mathbb {E}}(\widehat{\gamma _{\mu }})=\widehat{I({\widetilde{P}})}. \end{aligned}$$

Moreover, if \(\mu \) is ergodic, \(\gamma _{\mu }=I({\widetilde{P}})\) almost surely. In particular, \(I({\widetilde{P}})\) is positive definite.

We now proceed with the same strategy as in Sect. 4.3.1 to apply Strungaru’s criterion, and then conclude as in the beginning of Sect. 4.

Proposition 4.16

Assume the random perturbation \(X_{\xi }\) is stationary and weakly-mixing. Then almost surely, for every \(\lambda \in {\mathbb {R}}^d\) the diffraction measure \(\widehat{\gamma _{\xi }^{\lambda }}\) has no atoms.

Proof

This follows directly from Proposition 4.15 since the stationarity and the weak-mixing condition of \(X_{\xi }\) implies that \(\mu _{\xi }^{\lambda }\) is a stationary and weakly-mixing random measure for every \(\lambda \in {\mathbb {R}}^d\). The weak-mixing condition of \(\mu _{\xi }^{\lambda }\) and Proposition 4.15 implies by expanding the definition like in Lemma 4.11 that almost surely

$$\begin{aligned} \lim _{R\rightarrow \infty }\frac{1}{Vol(B_R)}|\gamma _{\xi }^{\lambda }(B_R)|=0, \end{aligned}$$

and therefore by Strungaru’s criterion (see [22, Proposition 4.1]), we get that \(\widehat{\gamma _{\xi }^{\lambda }}\) is almost surely continuous. Finally, the proof that this property holds for every \(\lambda \in {\mathbb {R}}^d\) follows along the same lines as the proof of Proposition 4.4, since for bounded f we have a control of the \(\lambda \)-gradients of

$$\begin{aligned} \frac{1}{Vol(B_R)}\int _{B_R}H_f(\mu _{\xi }^{\lambda }*\delta _{-t})dt \end{aligned}$$

as in (4.10). This finishes the proof of Proposition 4.16 and of Theorem C. \(\square \)

5.4.2 Some Examples of Stationary Random Perturbations

In this section we provide a class of examples of random perturbations of irreducible cut-and-project sets verifying the hypotheses of Theorem C.

Let \(L\subset {\mathbb {R}}^n\) be a lattice and denote by \(D_L\) its fundamental domain. Write \({\mathbb {R}}^n=V_{\textsf{phys}}\oplus V_{\textsf{int}}\), where \(V_{\textsf{phys}}\cong {\mathbb {R}}^d\) and \(V_{\textsf{int}}\cong {\mathbb {R}}^k\). We denote by \(\pi _{\textsf{phys}}\) and \(\pi _{\textsf{int}}\) the corresponding projections over \({\mathbb {R}}^d\) and \({\mathbb {R}}^k\) respectively. Fix \(W\subset V_{\textsf{int}}\) (called in the literature the window). The cut-and-project set \(X(L,W)\subset V_{\textsf{phys}}\) is defined by

$$\begin{aligned} X=X(L,W):=\pi _{\textsf{phys}}(L\cap \pi _{\textsf{int}}^{-1}(W)). \end{aligned}$$
(4.30)

We say that X is irreducible if the following conditions hold:

  1. (CP1)

    \(\pi _{\textsf{int}}(L)\) is dense in \(V_{\textsf{int}}\).

  2. (CP2)

    \(\pi _{\textsf{phys}}|_L\) is injective.

  3. (CP3)

    W is a bounded Borel measurable set with non-empty interior and such that its boundary has zero Lebesgue measure in \(V_{\textsf{int}}\).

It is known (see for instance [2, Theorem 9.4]) that under the conditions (CP1)-(CP3) the cut-and-project set (4.30) is a quasi-periodic set where (H.1) and (H.2) are satisfied automatically. Consider an L-stationary random process \((\xi '_q)_{q\in L}\) of identically distributed random vectors defined on \((\Omega ,{\mathcal {F}},{\mathbb {P}})\) with \(\xi '_q\in V_{\textsf{phys}}\), and let \(\eta \) be a random vector independent of \((\xi '_q)_{q\in L}\), which is distributed uniformly over \(D_L\). Then \(L_{\xi }^{\textsf{stat}}:=\{q+\xi '_q+\eta :\ q\in L\}\) is \({\mathbb {R}}^n\)-stationary.

Set \(X_\xi :=\pi _{\textsf{phys}}(L_\xi ^{\textsf{stat}}\cap (V_{\textsf{phys}}\times W))\). Note that identifying \(V_{\textsf{phys}}={\mathbb {R}}^d\), automatically \(X_\xi \) is \({\mathbb {R}}^d\)-stationary, since so is \(L_\xi ^{\textsf{stat}}\cap (V_{\textsf{phys}} \times W)\), as is direct to check. Also note that

$$\begin{aligned} L_\xi ^{\textsf{stat}}\cap (V_{\textsf{phys}}\times W) = \{q+\xi _q'+\eta :\ q\in L\cap (V_{\textsf{phys}}\times W-\eta )\}. \end{aligned}$$

Therefore, in order to have \(X_\xi =\{p+\xi _p:\ p\in X\}\) as required for Theorem C, we may set

$$\begin{aligned} \xi _p:=\xi _{\psi _\eta \circ \pi _{\textsf{phys}}^{-1}(p)}'+\pi _{\textsf{phys}}(\eta ), \end{aligned}$$

in which for each \(\eta \in D_L\) the map \(\psi _\eta : L\cap (V_{\textsf{phys}}\times W)\rightarrow L\cap (V_{\textsf{phys}}\times W-\eta )\) is a fixed bijection of bounded distortion. This is possible by assuming that X is up to bounded displacement (BD) equivalent to a lattice, which means that there is a bijection \(\phi : X\rightarrow {\mathcal {L}}\subset {\mathbb {R}}^d\) such that

$$\begin{aligned} \sup _{x\in X}|\phi (x)-x|<\infty . \end{aligned}$$

In fact, if \(\phi :X\rightarrow {\mathcal {L}}\) is a BD-equivalence from X to a lattice \({\mathcal {L}}\subset {\mathbb {R}}^d\), the conditions (CP1) and (CP3) imply that for every \(x\in {\mathbb {R}}^n\), there is a BD-equivalence \(\phi _{x}:\pi _{\textsf{phys}}(L\cap (V_{\textsf{phys}}\times W-x))\rightarrow {\mathcal {L}}\) (see [12, Proposition 2.6]). Hence by using (CP2) we can establish the desired bounded distortion bijection \(\psi _{\eta }:=\pi _{\textsf{phys}}^{-1}\circ \phi _{\eta }^{-1}\circ \phi \circ \pi _{\textsf{phys}}:L\cap (V_{\textsf{phys}}\times W)\rightarrow L\cap (V_{\textsf{phys}}\times W-\eta )\).

If we assume that \({\mathbb {E}}(|\xi '|^{d+\varepsilon })<\infty \) for some positive number \(\varepsilon \), then \({\mathbb {E}}|\xi |^{d+\varepsilon }<\infty \). Moreover, if \((\xi '_q)_{q\in L}\) is weakly-mixing (with respect to the natural shift in the lattice L), then the random perturbation \(X_{\xi }\) is weakly-mixing with respect to the \({\mathbb {R}}^d\)-translations. Thus \(X_{\xi }\) fulfils the hypotheses of Theorem C and therefore the irreducible cut-and-project set X is almost surely recoverable from \(X_{\xi }\).

6 Discussion and Open Problems

In this section we discuss the relations of our results with other notions, and we highlight two possible directions of future research.

6.1 “Quasi-stationarity” for the Recovery of General Meyer Sets

As already described in Sect. 4.3, recovery Theorems A, B hold also under weaker notions than the independence of the \(\xi _p\), and a good weakening is the hypothesis of stationarity and ergodicity. The drawback of this direction is that X needs to have a group structure, at least asymptotically, in order to define stationarity.

It seems natural to look for a definition of stationarity of the \((\xi _p)_{p\in X}\) that together with an associated notion of ergodicity, allows to obtain recovery Theorems A, B for more general quasi-periodic X, beyond asymptotically affine deformations of lattices. More precisely, if X is quasi-periodic then we may look for notions of invariance of the \((\xi _p)_{p\in X}\) in terms of the geometry of the almost-periods of X, which we could refer to as "quasi-stationarity" (however it seems to us that this term is already in use with a different meaning). Perhaps this can allow a more explicit recovery result for "quasi-stationary" \((\xi _p)_{p\in X}\), as opposed to Theorem C, in which stationarity under the \({\mathbb {R}}^d\)-action is imposed on the perturbed \(X_\xi \), and the relation between the geometry of X and that of the \((\xi _p)_{p\in X}\) is not explicitly quantified.

A natural setup where to start is that X satisfies the hypotheses (H.1) and (H.2), and that X is a Meyer set, which means that X is relatively dense and \(X-X:=\{p-q:\ p,q\in X\}\) is uniformly discrete.

If \(X-X\) is uniformly discrete then a first simplification occurs, since the limit defining \(\gamma _\xi ^\lambda (k)\) is trivially zero for \(k\notin X-X\). The difficulty is that, in general, for \(k\in X-X\) one can not ensure the existence and good control of the limits

$$\begin{aligned} \gamma _\xi ^\lambda (k)=\lim _{R\rightarrow \infty } \frac{1}{Vol(B_R)} \sum _{\begin{array}{c} p\in X\cap B_R\\ p-k\in X \end{array}}\mu _\xi ^\lambda (p)\overline{\mu _\xi ^\lambda (p-k)}. \end{aligned}$$
(5.1)

In this case, the main open question is to find explicit further conditions that we need to impose on X and on the \(\xi _p\) in order to achieve the result of Proposition 4.4.

6.2 Recovering the Structure Factor and Almost-Sure Recovery of the Diffraction Measure

In the setting of Theorem A, if the characteristic function \(\varphi (\lambda ):={\mathbb {E}}[e^{-2\pi i\langle \xi , \lambda \rangle }]\) vanishes at some points of the support of \({{\widehat{\delta }}}_X\), this means that this part of the spectrum of X is not recoverable, a phenomenon called cloaking of X by the random perturbations \(\xi _p\).

In [16] the authors present a result analogous to our theorems, but in expectation form, formulated in terms of structure factors. In our notation, the structure factor of a random discrete set \(X_\xi =\{p+\xi _p:\ p\in X\}\) is the function \(S_{X_\xi }:{\mathbb {R}}^d\rightarrow [0,+\infty ]\) formally given by

$$\begin{aligned} S_{X_\xi }(\lambda ):=\lim _{R\rightarrow \infty }\mathbb E\left[ \frac{1}{\#(X\cap B_R)}\left| \sum _{p\in X\cap B_R}e^{-2\pi i \langle p+\xi _p, \lambda \rangle }\right| ^2\right] =\frac{\mathbb E[{\widehat{\gamma }}_{X_\xi }(\lambda )]}{\textrm{dens}(X)}, \end{aligned}$$
(5.2)

where \(\gamma _{X_\xi }(\lambda )\) is the autocorrelation of \(X_{\xi }\). Note that the above limit may not exist, or may be defined only in a weak (distributional) sense, if \(X_\xi \) is not regular enough.

The above quantity is most commonly studied for the case that X is not deterministic, but has a well-defined density, and \(\textrm{dens}(X)=\textrm{dens}(X_\xi )={{\widehat{\gamma }}}_{X_\xi }(0)\) almost surely. Often X is assumed to be a stationary spatial point process, the simplest example being a "stationarized lattice", i.e. X is lattice translated by a random vector \(\tau \) uniformly distributed over its fundamental domain. In this case, if the \(\xi _p\) are assumed to be i.i.d. and independent of (the random process) X, then as done in [16, App. A] one can rewrite the square modulus from (5.2) as a double sum, and then split the expectation as a product of expectations expressed in terms of \(X, \xi _p, p\in X\). Thus we get directly (where the \(-1\) terms come from the diagonal term in the double sums obtained expanding the square in (5.2))

$$\begin{aligned} S_{X_\xi }(\lambda ) - 1 = \left| {\mathbb {E}}\left[ e^{-2\pi i\langle \xi , \lambda \rangle }\right] \right| ^2 \left( S_X(\lambda ) - 1\right) . \end{aligned}$$
(5.3)

Our Theorems A, B, C give conditions for the almost-sure recovery of \(\widehat{\delta _X}\), the Fourier transform of the empirical measure of the set X. The same strategy does not directly give the almost-sure recovery of the diffraction measure of X, which would be a strengthening of (5.3), and it would be interesting to continue our study in that direction. More precisely, this requires to establish conditions on \(X,(\xi _p)_{p\in X}\) such that a limit like the below is ensured almost surely for all \(\lambda \in {\mathbb {R}}^d\):

$$\begin{aligned} \lim _{R\rightarrow \infty }\frac{1}{|B_R|}\sum _{\begin{array}{c} p,q\in X\cap B_R \end{array}}e^{-2\pi i\langle p-q,\lambda \rangle }\left( e^{-2\pi i\langle \xi _p-\xi _q,\lambda \rangle }-\left| {\mathbb {E}}[e^{-2\pi i\langle \xi ,\lambda \rangle }]\right| ^2\right) =0. \end{aligned}$$

If we prove the above, then this would substitute Proposition 4.2, and would imply by the same reasoning as in Sect. 4 that almost surely for all \(\lambda \)

$$\begin{aligned} \widehat{\gamma _{X_\xi }}(\lambda ) - \textsf{dens}(X)= \left| \mathbb E\left[ e^{-2\pi i\langle \xi , \lambda \rangle }\right] \right| ^2 \left( \widehat{\gamma _X}(\lambda )-\textsf{dens}(X)\right) , \end{aligned}$$
(5.4)

which is the almost-sure version of (5.3) and which already appears in [13] under different hypotheses over the set X. In fact, in [13] it is assumed that X is an uniformly discrete set with asymptotic density and satisfying an ergodicity condition. Hof’s ergodicity hypotheses are not directly comparable to our (H1), (H2). We leave to future work the elucidation of implications between hypotheses of ergodicity of \(\delta _X\) like those in [13] and hypotheses on \({\widehat{\gamma }}_X\) such as (H2).