2.1 Cut and project schemes

We briefly review the definition of cut and project schemes and properties of weak model sets. For more details, we recommend [Reference Baake and Grimm1, Reference Baake and Grimm2, Reference Lenz and Richard19, Reference Moody and Moody23, Reference Moody, Axel, Dénoyer and Gazeau24, Reference Richard and Strungaru27, Reference Strungaru, Baake and Grimm35, Reference Strungaru36], just to name a few.

Given a CPS $(G,H, {\mathcal L})$ , we denote by $\mathbb T:=(G \times H) /{\mathcal L}$ and by $L:= \pi _G({\mathcal L})$ . There exists a mapping, call the star mapping, $\star : L \to H$ so that ${\mathcal L}= \{ (x,x^\star ) : x \in L \}$ . Next, for each $W \subseteq H$ , we can define

We are interested in , when W is a precompact Borel set. For such set $W \subseteq H$ , we define its covariogram function $c_W : H \to {\mathbb C}$ via $c_W(t):= \theta _{H}(W \cap (t+W))$ .

More generally, for each $h : H \to {\mathbb C}$ , we can define a formal sum

$$\begin{align*}\omega_h:= \sum_{x \in L} h(x^\star) \delta_x. \end{align*}$$

If h is compactly supported and bounded, then $\omega _h$ is a measure. There are many other conditions on h which ensure that $\omega _h$ is a measure; see, for example, [Reference Lenz and Richard19, Reference Richard and Strungaru27, Reference Strungaru36, Reference Strungaru37].

Note here that, for all $s \in L$ , we have

(2.1)

(2.2) $$ \begin{align} \tau_s\omega_h &=\sum_{x \in L} h(x^\star) \delta_{s+x} =\sum_{y \in L} h(y^\star-s^\star) \delta_{y}= \omega_{\tau_{s^\star}h}. \end{align} $$

We will often use these two formulas.

Given a CPS $(G,H, {\mathcal L})$ , we can define

$$\begin{align*}{\mathcal L}^0:= \{ (\chi, \psi) \in \widehat{G} \times \widehat{H} : \chi(x)\psi(x^\star)=1 \forall x \in L \}. \end{align*}$$

Then, $(\widehat {G}, \widehat {H}, {\mathcal L}^0)$ is a CPS[Reference Moody and Moody23], called the CPS dual to $(G, H, {\mathcal L})$ . This dual CPS plays the central role in the study of diffraction of model sets.

To avoid confusion, for each $h : \widehat {H} \to {\mathbb C}$ , we will denote the comb given by h in the dual CPS $(\widehat {G}, \widehat {H}, {\mathcal L}^0)$ by $\omega ^\star _{h}$ , that is,

$$\begin{align*}\omega_{h}^\star:= \sum_{(\chi, \chi^\star) \in {\mathcal L}^0} h(\chi^\star) \delta_{\chi}. \end{align*}$$

When dealing with functions, one can use Følner sequences to calculate averages (means). Below, we will often calculate the average (density) of point sets, and in this situation, Følner sets do not suffice. Indeed, the sequence

$$\begin{align*}F_n=[-n-\frac{1}{2}, n+\frac{1}{2}] \backslash \cup_{k=-n}^n [k-\frac{1}{2^n}, k+\frac{1}{2^n}] \,, \end{align*}$$

which is the interval $[-n-\frac {1}{2}, n+\frac {1}{2}]$ with small holes around the integers, is a Følner sequence which cannot be used to average the Delone set ${\mathbb Z}$ , or weighted Dirac combs on ${\mathbb Z}$ . Similarly, for each Delone subset $\Lambda \subseteq {\mathbb R}$ , it is easy to construct a Følner sequence which has an empty intersection with $\Lambda $ .

As it was observed by Schlottmann [Reference Schlottmann, Baake and Moody29], to avoid this issue when dealing with point sets or measures, one needs to work instead with van Hove sequences (see [Reference Lenz, Spindeler and Strungaru20, Definition 1.6] for definition and properties). Any van Hove sequence is a Følner sequence, and any $\sigma $ -compact LCAG G admits a (tempered) van Hove sequence.

Given a uniformly discrete set $\Lambda \subseteq G$ and a van Hove sequence ${\mathcal A}=\{A_k\}$ , we define

$$ \begin{align*} \underline{\mathrm{dens}}_{{\mathcal A}}(\Lambda) &:= \liminf_k \frac{\mathrm{card}(\Lambda \cap A_k)}{|A_k|} \\ \overline{\mathrm{dens}}_{{\mathcal A}}(\Lambda) &:= \limsup_k \frac{\mathrm{card}(\Lambda \cap A_k)}{|A_k|}. \end{align*} $$

If the following limit exists, or equivalently if $\underline {\mathrm {dens}}_{{\mathcal A}}(\Lambda )=\overline {\mathrm {dens}}_{{\mathcal A}}(\Lambda )$ , we define

$$\begin{align*}\mathrm{dens}_{{\mathcal A}}(\Lambda):= \lim_k \frac{\mathrm{card}(\Lambda \cap A_k)}{|A_k|}. \end{align*}$$

Let us next recall the following result of [Reference Huck and Richard12].

Theorem 2.1 [Reference Huck and Richard12]

Let $(G,H, {\mathcal L})$ be a CPS and $W \subseteq H$ be a precompact. Then, for all van Hove sequences ${\mathcal A}$ , we have

$$ \begin{align*} \mathrm{dens}({\mathcal L}) \theta_H(W^\circ) \leq \underline{\mathrm{dens}}( \curlywedge (W)) \leq \overline{\mathrm{dens}}( \curlywedge (W)) \leq \mathrm{dens}({\mathcal L}) \theta_H(\overline{W}). \end{align*} $$

We will use the following definitions.

Let us complete this section by recalling the following result from [Reference Moody25].

Theorem 2.3 [Reference Moody25, Theorem 1]

Let $(G,H, {\mathcal L})$ be a CPS, let $W \subseteq H$ be a measurable precompact set, and let ${\mathcal A}$ be a tempered van Hove sequence in G. Then, for almost all $(s,t)+{\mathcal L} \in (G \times H)/ {\mathcal L} =: \mathbb T$ , we haveFootnote ¹

(2.3)

2.2 Dynamical systems

Next, we review few basic facts about dynamical systems of translation bounded measures. Let us start by recalling the following result of [Reference Spindeler and Strungaru32] (compare [Reference Baake and Lenz5]).

Lemma 2.4 [Reference Spindeler and Strungaru32]

Let $\mathbb X \subseteq {\mathcal M}^\infty (G)$ be any G invariant vaguely closed set. Then, $\mathbb X$ is vaguely compact if and only if there exists some C and an open precompact set $\emptyset \neq U \subseteq G$ such that

$$\begin{align*}\mathbb X \subseteq \{ \mu \in {\mathcal M}^\infty(G): \| \mu \|_{U} \leq C \} =: {\mathcal M}_{C,U}(G). \end{align*}$$

Moreover, in this case, the vague topology on $\mathbb X$ is metrizable.

Next, let us review the following definition, which is inspired by the general Birkhoff ergodic theorem for continuous group actions [Reference Chatard7, Reference Tempelman38].

Definition 2.3 Let $(X,G,\mathfrak { m})$ be an ergodic dynamical system and let ${\mathcal A}$ be a van Hove sequence. An element $x \in X$ is called generic for $\mathfrak { m}$ if

$$ \begin{align*} \lim_{k} \frac{1}{|A_k|} \int_{A_k} f(\tau_t x) \mathrm{d} t = \int_{X} f(y) \mathrm{d} \mathfrak{ m}(y) \end{align*} $$

holds for all $f \in C(X)$ .

We will need the following simple characterizations of generic elements. This has been used implicitly in [Reference Baake and Lenz5, Reference Lenz, Spindeler and Strungaru20, Reference Lenz, Spindeler and Strungaru21] just to name a few. Since the result is well known and trivial to prove, we skip the proof.

Proposition 2.5 Let $(X, G, \mathfrak { m})$ be an ergodic dynamical system. Then, $x \in X$ is generic for $\mathfrak { m}$ if and only if the set of $f \in C(X)$ for which

$$\begin{align*}\int_{X} f(y) \mathrm{d} \mathfrak{ m}(y)= \lim_k \frac{1}{|A_k|} \int_{A_k} f(\tau_tx) \mathrm{d} t \end{align*}$$

is dense in $C(X)$ .

By combining this result with Stone–Weierstrass theorem, we immediately get the following proposition.

Proposition 2.6 Let $\mathbb X \subseteq {\mathcal M}_{C,U}$ be vaguely compact and G-invariant, and let $\mathfrak { m}$ be an ergodic measure on $\mathbb X$ . If all measures in $\mathbb X$ are real valued, then a measure $\mu \in \mathbb X$ is generic for $\mathfrak { m}$ if and only if, for all $n \in \mathbb N$ and $\varphi _1,\ldots , \varphi _n \in C_{\text {c}}(G)$ , we have

(2.4) $$ \begin{align} &\lim_{k} \frac{1}{|A_k|} \int_{A_k} \mu*\varphi_1(t) \mu*\varphi_2(t) \cdot \dots \cdot \mu*\varphi_n(t) \mathrm{d} t \nonumber \\ & \quad = \int_{\mathbb X(B)} f_{\varphi_1}(\omega) f_{\varphi_2}(\omega) \cdot \dots \cdot f_{\varphi_n}(\omega) \mathrm{d} \mathfrak{ m}(\omega). \end{align} $$

2.3 The hull and extended hull

Recall that, for a precompact open set $\emptyset \neq U \neq \subseteq G$ , we denote by $\mathcal {D}_U$ the set of U-uniformly discrete subsets of G. The mapping $\Lambda \to \delta _{\Lambda }$ induces an embedding of $\mathcal {D}_U$ into ${\mathcal M}_{1, U}$ with compact range. We can then equip $\mathcal {D}_U$ with the local rubber topology $\tau _{lr}$ , which is the pullback through this mapping of the vague topology.

Now, let $(G, H, {\mathcal L})$ be a cut and project scheme and $B \subseteq H$ be any precompact set. Then, there exists some precompact open set $\emptyset \neq U \neq \subseteq G$ such that, for all $(s,t) \in G \times H$ , we have .

Definition 2.4 The hull of and extended hull ${\mathcal M}_B^G$ of B are defined as

where the closure is taken in the local rubber topology.

Remark 2.7 As observed by Schlottmann [Reference Schlottmann, Baake and Moody29], we get a natural G-mapping

Moody’s uniform distribution result, Theorem 2.3, implies that almost surely almost all positions $(s,t)+{\mathcal L} \in G \times H / {\mathcal L}$ give us point sets whose distributional average in G matches the corresponding integrals over the window. Since we are interested in those elements, and since the original position may not be one of those generic positions, one needs to work with the extended hull ${\mathcal M}_B^G$ and not just the classical hull .

For regular model sets, we have , but this is in general not true. Indeed, consider the CPS $({\mathbb R}, H, {\mathcal L})$ and the window W of the visible points of the lattice [Reference Baake, Huck and Strungaru4]. Then, there exists positions $t \in H$ such that $(t+W) \cap L^\star =\emptyset $ . Pick one such position t and set $B=t+W$ . Then, while the extended hull ${\mathcal M}_B^G$ contains the visible points of the lattice.

From a dynamical point of view, the above mapping can be seen as a (generally) discontinuous observation introduced by the window on the compact group $\mathbb T$ . We will show below that there exists a natural measure (the Mirsky measure) which describes the distribution generated by the observation, and that almost all $(s,t)+{\mathcal L} \in \mathbb T$ define observations which are generic for this measure.

We complete this section by proving the following simple lemma.

Proof “(a)” Since $L^\star $ is dense in H, we can find a precompact neighborhood $t \in V$ and a net $t_{\alpha }$ with $t_\alpha \in V$ which converges in H to t. Then, using the uniform continuity of h, we get that

(2.5) $$ \begin{align} \lim_\alpha \| \tau_{t_\alpha}h - \tau_{t}h \|_\infty =0. \end{align} $$

Next, all these functions are supported in the compact set $\overline {V}+\mathrm {supp}(h)$ . Therefore, all the measures $\omega _{ \tau _{t_\alpha }h}, \omega _{ \tau _{t}h}$ are supported inside the uniformly discrete set . Moreover, (2.5) implies that for all , we have

$$\begin{align*}\omega_{ \tau_{t}h}( \{x \})= \lim_\alpha \omega_{ \tau_{t_\alpha}h}( \{x \}). \end{align*}$$

This combined with the equi uniform discreteness of their support implies that $\omega _{ \tau _{t_\alpha }h}$ converges vaguely to $\omega _{ \tau _{t}h}$ .

Finally, since $t_\alpha \in L^\star $ , there exists some $s_\alpha \in L$ such that $t_\alpha = s_\alpha ^\star $ . Then, by (2.2), we have

$$\begin{align*}\omega_{ \tau_{t_\alpha}h}=\tau_{s_\alpha} \omega_h \in \mathbb X(\omega_h). \end{align*}$$

The claim follows.

“(b)” follows immediately from (a) and the unique ergodic theorem.

2.4 Besicovitch almost periodicity

Consider a Følner sequence ${\mathcal A}$ on G. On the space of a locally bounded measurable function, we can then define a semi-norm, called the Besicovitch p-seminorm for functions

$$\begin{align*}\|f \|_{{\textsf{B}},p,{\mathcal A}}:= \left( \limsup_k \frac{1}{|A_k|}\int_{A_k} \left|f(t) \right|^p\mathrm{d} t \right)^{\frac{1}{p}}. \end{align*}$$

With this in mind, let us recall the following definition.

Definition 2.5 A function $f \in L^1_{loc}(G)$ is called Besicovitch $1$ -almost periodic if for each $\varepsilon>0$ there exists a Bohr almost periodic function g such that

$$\begin{align*}\| f-g \|_{{\textsf{B}},1,{\mathcal A}} < \varepsilon. \end{align*}$$

A translation bounded measure $\mu $ is called Besicovitch almost periodic if $\mu *\varphi $ is a Besicovitch $1$ -almost periodic function for all $\varphi \in C_{\textsf {c}}(G)$ .

The space of Besicovitch almost periodic measures is denoted by $\mathcal {AP}\text {-}{Bes}_{{\mathcal A}}(G)$ .

In this article, we will be interested in point sets $\Lambda $ for which $\delta _{\Lambda } \in \mathcal {AP}\text {-}{Bes}_{{\mathcal A}}(G)$ . Because of this, we introduce the following notation:

$$\begin{align*}\mathcal{AP}\text{-}{Bes}_{{\mathcal A}, ps}(G):= \{ \Lambda \subseteq G : \delta_{\Lambda} \in \mathcal{AP}\text{-}{Bes}_{{\mathcal A}}(G) \}. \end{align*}$$

Below, we will make use of the following two results.

For a measure $\mu \in {\mathcal M}^\infty $ , we will denote by

$$\begin{align*}\|\mu \|_{{\textsf{B}},{\mathcal A}}:= \limsup_k \frac{\left| \mu \right|(A_k)}{|A_k|}. \end{align*}$$

For all $\varphi \in C_{\text {c}}(G)$ and $\mu \in {\mathcal M}^\infty (G)$ , we have [Reference Lenz, Spindeler and Strungaru20, Lemma 2.13]

(2.6) $$ \begin{align} \| \varphi*\mu \|_{{\text{B}},1,{\mathcal A}} \leq \| \mu \|_{{\text{B}},{\mathcal A}} \| \varphi \|_1. \end{align} $$

4.1 Construction of $\boldsymbol{U_n, K_n,}$ and $\boldsymbol{g_n}$

For each n, by [Reference Rudin28, Theorem 2.14(c) and (d)], there exists some compact set $K_n$ and open precompact set $U_n \subseteq U$ such that

$$\begin{align*}K_n \subseteq B \subseteq U_n \, \text{ and } \theta_{H} (U_n \backslash K_n) < \frac{1}{n}. \end{align*}$$

We fix one such choice of $K_n$ and $U_n$ for each n and, by eventually replacing $K_n$ by $K_1 \cup K_2 \cup \cdots \cup K_n$ and $U_n$ by $U_1 \cap U_2 \cap \cdots \cap U_n$ , we can assume without loss of generality that

$$\begin{align*}K_1 \subseteq K_2 \subseteq \cdots \subseteq K_ n \subseteq \cdots \subseteq B \subseteq \cdots \subseteq U_n \subseteq \cdots \subseteq U_2 \subseteq U_1. \end{align*}$$

Next, for each n, fix some $g_n \in C_{\textsf {c}}(G)$ such that $1_{K_n} \leq g_n \leq 1_{U_n}$ . The sets $K_n, U_n$ and the functions $g_n$ are fixed for the remaining of the section.

We can now introduce the following definition.

Definition 4.1 We denote by

and we will refer to this as the set of fb-generic points.

Lemma 4.2 Let $(s,t)+{\mathcal L} \in \mathbb T$ . Then, $(s,t)+{\mathcal L} \in \mathcal {MG}(B)$ if and only if, for all $\varphi \in C_{\textsf {c}}(G)$ , we have

Proof $\Longrightarrow $ follows from (2.6).

$\Longleftarrow $ We know that which is uniformly discrete. Let $V \subseteq G$ be any open set such that is V-uniformly discrete. Then, is $v+V$ -uniformly discrete for all $v \in G$ .

Let $C:= \{ \varphi \in C_{\textsf {c}}(G) : \exists v \in G \text { s.t. } \mathrm {supp}(\varphi )\subseteq v+V \}$ . Strungaru [Reference Strungaru, Baake and Grimm35, Lemma 5.8.3] implies that for all $\varphi \in C$ , we have

This immediately implies that

for all $\varphi \in C$ . As $\text {Span}(C) =C_{\textsf {c}}(G)$ , the claim follows.

4.2 The main result

For each n, define

and let

(4.1) $$ \begin{align} {\mathcal T}:= \bigcap_n (X_n \cap Y_n). \end{align} $$

By Theorem 2.3, $X_n, Y_n \subseteq \mathbb T$ contain subsets of full measure in $\mathbb T$ . In particular, since we assume the Haar measure to be complete, $X_n$ and $Y_n$ have full measure in $\mathbb T$ . It follows that $\theta _{\mathbb T}({\mathcal T}) =1$ . Note here that $(s,t) +{\mathcal L} \in {\mathcal T}$ if and only if for all n, we have

(4.2)

(4.3)

We can now prove the following result (compare [Reference Keller, Richard and Strungaru16, Theorem 4.6]).

Proof “(a)” Let $(s,t)+{\mathcal L} \in {\mathcal T}$ . We have

which together with (4.2) and (4.3) gives

Thus,

which proves (a).

“(b)” Let $(s,t) +{\mathcal L} \in \mathcal {MG}(B)$ . Then, (2.6) gives that for all $\varphi \in C_{\textsf {c}}(G)$ , we have

(4.4)

Since $ \tau _{-s}\omega _{\tau _{t}g_n} \in \mathcal {AP}\text {-}{Strong} (G)$ , (b) follows.

“(c,d,e,f,g)” Let $(s,t) +{\mathcal L} \in \mathcal {MG}(B)$ . Since , for each character $\chi $ , the Fourier–Bohr coefficient exists by [Reference Lenz, Spindeler and Strungaru20, Proposition 4.32].

Moreover, for all $\varphi \in C_{\textsf {c}}(G)$ , we have

with the second inequality following from (4.4) the inequality $\left |a_\chi ^{\mathcal A}(f)\right | \leq \|f\|_{{\textsf {B}},1,{\mathcal A}}$ . This shows that

(4.5)

Now, when $\chi =0$ is the trivial character, (4.5) yields (c).

Next, since , the autocorrelation $\gamma $ of exists with respect to ${\mathcal A}$ , is strongly almost periodic, and satisfies [Reference Lenz, Spindeler and Strungaru20, Theorem 4.39].

which is (g).

Moreover, since

$$\begin{align*}a_\chi^{\mathcal A}( \tau_{-s}\omega_{\tau_{t}g_n})= \left\{ \begin{array}{@{}cc} \mathrm{dens}({\mathcal L}) (\chi(s)\chi^\star(t) ) \int_{H} g_n(r) \chi^\star(r) \mathrm{d} r & \text{ if } \chi \in \pi_{\widehat{G}}({\mathcal L}^0) \\ 0 & \text{ otherwise } \end{array} \right.\!\!\!, \end{align*}$$

(f) follows. Moreover, combining (f) with the fact that $\gamma $ is strongly almost periodic and satisfies the CPP, we get (e).

Finally, the measure $\eta := \mathrm {dens}({\mathcal L}) \omega _{c(B)}$ is positive definite and hence Fourier transformable. Moreover,

$$\begin{align*}\widehat{\eta}=(\mathrm{dens}({\mathcal L}))^2 \omega^\star_{|\check{1_B}|^2}= \widehat{\gamma} \end{align*}$$

with the second equality following from (e). This yields (d).

By combining Theorem 4.3 with Theorem 2.10, we get the following corollary.

In the next section, we are going to prove that $\mathfrak { m}_{(s,t)+{\mathcal L}}$ does not depend on the choice of $(s,t)+{\mathcal L} \in \mathcal {MG}(B)$ .

4.3 The Mirsky measure

We can now introduce the Mirsky measure associated with the window B. Since $\omega _{g_n}$ are strongly almost periodic, the hull $\mathbb X(\omega _{g_n} )$ is uniquely ergodic. Let $\mathfrak { m}_n$ be the unique ergodic measure on $\mathbb X(\omega _{g_n} )$ . We will show that for all $(s,t)+{\mathcal L} \in \mathcal {MG}(B)$ , the sequence $\mathfrak { m}_n$ converges vaguely to $\mathfrak { m}_{(s,t)+{\mathcal L}}$ .

By construction, all measures $\omega _n$ are supported inside the model set . It follows immediately that there exists some open precompact set U and some $C>0$ such that all the hulls $\mathbb X(\omega _n)$ and ${\mathcal M}_B^G$ are subsets of ${\mathcal M}_{C,U}$ . This set is G-invariant and vaguely compact [Reference Baake and Lenz5]. Let $\mathbb Y$ denote the set of real valued measures in ${\mathcal M}_{C,K}$ . Then, $\mathbb Y$ is G invariant, vaguely compact, and contains $\mathbb X(g_n)$ and ${\mathcal M}_B^G$ . Also, we can consider the measures $\mathfrak { m}_n$ and $\mathfrak { m}_{(s,t)+{\mathcal L}}$ as measures on $\mathbb Y$ .

As usual, for $\phi \in C_{\textsf {c}}(G)$ , we define $F_{\phi } : \mathbb Y \to {\mathbb C}$ via

$$\begin{align*}F_{\phi}(\omega)=\omega*\phi(0). \end{align*}$$

Since $\mathbb Y$ only consists of real valued measures, we have $\overline {F_{\phi }}= F_{\overline {\phi }}$ .

Let us first prove the following preliminary lemma.

Lemma 4.5 Let $\phi _1, \ldots , \phi _j \in C_{\textsf {c}}(G)$ . Then, for all $(s,t)+{\mathcal L} \in \mathcal {MG}(B)$ , we have

Proof The limit

exists by the Besicovitch almost periodicity of . A short computation shows

where

Therefore, the definition of $\mathcal {MG}(B)$ and (2.6) give

Finally, by Lemma 2.8 and the unique ergodicity of $m_n$ , we have

$$\begin{align*}\lim_k \frac{1}{|A_k|} \int_{A_k} \prod_{i=1}^j \phi_i*(\tau_{-s}\omega_{\tau_{t}g_n}) \mathrm{d} t= \int_{{\mathcal M}_{C,K}}\prod_{i=1}^j F_{\phi_i} (\omega) \mathrm{d} \mathfrak{ m}_n (\omega) \end{align*}$$

which proves the claim.

We can now prove that all $m_{(s,t)+{\mathcal L}}$ are the same.

Proof Fix an arbitrary $(s,t) +{\mathcal L} \in \mathcal {MG}(B)$

Consider the algebra $\mathbb A$ generated by $\{ F_{\phi }: \phi \in C_{\textsf {c}}(G) \} \cup \{ 1_{\mathbb Y} \}$ , which is dense in $C(\mathbb Y)$ .

Since $ (s,t)+ {\mathcal L} \in \mathcal {MG}(B)$ , the measure is generic for $\mathfrak { m}_{(s,t)+{\mathcal L}}$ and hence, for all $\varphi _1, \ldots , \varphi _j \in C_{\textsf {c}}(G)$ , we have

(4.6)

This together with Lemma 4.5 gives that for all $\varphi _1, \ldots , \varphi _j \in C_{\textsf {c}}(G)$ , we have

$$\begin{align*}\mathfrak{ m}_{(s,t)+{\mathcal L}} (\prod_{i=1}^j F_{\phi_i}) = \lim_n \mathfrak{ m}_{n} (\prod_{i=1}^j F_{\phi_i}). \end{align*}$$

Now, since $\mathfrak { m}_{(s,t)+{\mathcal L}}(1_{\mathbb Y})=1=\mathfrak { m}_n(1_{\mathbb Y})$ , it follows immediately that

(4.7) $$ \begin{align} \mathfrak{ m}_{(s,t)+{\mathcal L}}(F)= \lim_n \mathfrak{ m}_n(F) \qquad \forall F \in \mathbb A. \end{align} $$

Since G is second countable and $\mathbb Y$ is metrizable and compact, the set $\text {Prob}(\mathbb Y)$ of probability measures on $\mathbb Y$ is vaguely compact and metrizable. Therefore, to show that $m_n$ converges vaguely to $\mathfrak { m}_{(s,t)+{\mathcal L}}$ , it suffices to show that $\mathfrak { m}_{(s,t)+{\mathcal L}}$ is the unique vague cluster point of $\mathfrak { m}_n$ . Let $\mathfrak { m}$ be any vague cluster point of $\mathfrak { m}_n$ . Then, (4.7) yields

$$\begin{align*}\mathfrak{ m}_{(s,t)+{\mathcal L}}(F)= \lim_n \mathfrak{ m}_n(F) \qquad \forall F \in \mathbb A. \end{align*}$$

Therefore, the denseness of $\mathbb A$ gives $\mathfrak { m}_{(s,t)+{\mathcal L}}=\mathfrak { m}$ , completing the proof.

This leads to the next definition, compare [Reference Keller, Richard and Strungaru16].

Definition 4.2 We define the Mirsky measure $Q_{B}^G$ of B as

$$\begin{align*}Q_{B}^G= \mathfrak{ m}_{(s,t)+{\mathcal L}} \qquad \forall (s,t)+{\mathcal L} \in \mathcal{MG}(B). \end{align*}$$

By combining Theorem 4.3, Proposition 4.6, and [Reference Lenz, Spindeler and Strungaru20, Reference Lenz, Spindeler and Strungaru21], we get the following proposition.

We complete the section by characterizing $\mathcal {MG}(B)$ in terms of genericity with respect to the Mirsky measure and Fourier–Bohr coefficients.

Lemma 4.8 Let $(s,t) +{\mathcal L} \in \mathbb T$ . Then, $(s,t) +{\mathcal L} \in \mathcal {MG}(B)$ if and only if and

In particular, $\mathcal {MG}(B)$ does not depend on the choice of $U_n, K_n$ , and $g_n$ .

Proof $\Longrightarrow $ follows from Theorem 4.3.

$\Longleftarrow $ Let $0 \in V \subseteq G$ be open such that is V-uniformly discrete. Pick some $\varphi \in C_{\textsf {c}}(G)$ such that $\varphi \geq 0, \int \varphi =1$ , and $\mathrm {supp}(\varphi ) \subseteq V$ .

Since is Besicovitch almost periodic, we have by the Parseval identity

with the last equality following from the Poisson summation formula [Reference Richard and Strungaru27].

Note here that $\left ( (\varphi ^\dagger )*\widetilde {(\varphi ^\dagger )} \otimes (1_B -g_m)*\widetilde {(1_B-g_m)} \right )$ are supported inside a fixed compact set, and hence $ \delta _{{\mathcal L}} \left ( (\varphi ^\dagger )*\widetilde {(\varphi ^\dagger )} \otimes (1_B -g_m)*\widetilde {(1_B-g_m)} \right )$ is a finite sum. Since $\left ( (\varphi ^\dagger )*\widetilde {(\varphi ^\dagger )} \otimes (1_B -g_m)*\widetilde {(1_B-g_m)} \right )$ converge point wisely to $0$ , we get

and hence by [Reference Lenz, Spindeler and Strungaru20, Lemma 2.7], we have

Finally, the choice of $\varphi $ and [Reference Strungaru, Baake and Grimm35, Lemma 5.8.3] give the claim.