1 Introduction and overview↩︎

The set of square-free integers induces an interesting topological dynamical system (TDS) with pure-point dynamical spectrum relative to the patch frequency (or Mirsky) measure. Likewise, the visible (or primitive) points of the integer lattice \(\mathbb{Z}^2\) induce a TDS with analogous properties. Both systems have many generalisations, either in the direction of \(\mathcal{B}\)-free systems in one dimension or towards their analogues among algebraic \(\mathcal{B}\)-free lattice systems in higher dimensions. An interesting class of examples emerges from considering \(\mathcal{B}\)-free systems of integers in algebraic number fields, as well as certain \(\mathcal{B}\)-free lattice systems in the sense of [1].

Some of these systems are re-analysed here from a spectral perspective, where we employ the cut and project formalism for weak model sets developed in [2], [3]. To do so, we use the notion of a locally compact Abelian group (LCAG) and refer to [4] for background on harmonic analysis in this setting. Our motivation comes from the Halmos–von Neumann theorem for ergodic dynamical systems with pure-point spectrum. It states that two such systems are measure-theoretically isomorphic if and only if they have the same spectrum, thus establishing one of the first cases of a complete invariant in dynamical systems theory.

The pure-point nature of the systems analysed below can be obtained as a consequence of identifying them as weak model sets of maximal density and then using a key result from [5]. To actually profit from spectral information, we need to calculate the spectrum explicitly, which often is more difficult than merely establishing the pure-point nature of a system. The point of this paper is that large and interesting families of number-theoretic dynamical systems exist where this is actually possible, thus giving an explicit group-theoretic invariant to tell them apart with respect to measure-theoretic conjugacy.

The systems we are interested in have many invariant measures. Two among them are special, namely the natural patch frequency measure, which is also known as the Mirsky measure, and the measure of maximal entropy. With respect to the Mirsky measure, which has zero entropy, our systems have pure-point dynamical spectrum, but trivial topological point spectrum. Nevertheless, via a generic element \(V\), we can calculate the eigenfunctions in closed form via the Fourier–Bohr (FB) coefficients of \(V\). The eigenfunctions are known to be continuous on a set of full measure [5], [6].

The measure of maximal entropy emerges from a joining (in fact, a direct product) of the Mirsky measure with the Bernoulli measure of the fair coin toss, and has mixed spectrum with a pure-point part (and eigenfunctions that also derive from the FB coefficients of generic elements) and an absolutely continuous part, with countable Lebesgue spectrum. We give an easy and constructive proof for the eigenfunctions with respect to the measure of maximal entropy, and otherwise concentrate on the spectral theory with respect to the Mirsky measure.

The paper is organised as follows. First, in Section 2, where we also introduce our notation, we summarise key results of the visible lattice points and the corresponding dynamical system in Theorem 1 and Corollary 2, which will serve as our guiding example. We introduce a simplified method for the computation of the spectrum in Remark 3, which will be vital for all later examples. We then construct the eigenfunctions via the FB coefficients in Theorem 5 and derive the corresponding situation for the measure of maximal entropy in Corollary 7, via an application of Kolmogorov’s strong law of large numbers.

An analogous situation is then met for the power-free integers in Section 3, as summarised in Theorem 8, which suggests that a more general setting is reasonable and possible. This is derived in Section 4 in the setting of \(\mathcal{B}\)-free lattice systems of Erdős type, which can be seen as a general framework for our approach. Here, the spectrum and its group structure is presented in Proposition 10 and Theorem 11, thus setting our general scene.

Then, we compute more concrete results for power-free integers in quadratic number fields in Section 5, where we treat the cases of imaginary and real fields separately. The spectral results are given in Theorems 16 and 18, respectively. This section requires some ideal-theoretic arguments, though we begin both parts with cases of class number \(1\) that permit working with numbers rather than ideals.

Some underlying material on uniform distribution, which we need for computing the eigenfunctions, is provided in the Appendix.

2 A guiding example with many facets↩︎

Let us start with a well-studied example that will be a guide for most of our generalisations. It emerges from the set of visible points \(V\) of the integer lattice \(\mathbb{Z}^d\), defined by \[V \, \mathrel{\mathop{:}}=\, \bigl\{ (x^{\phantom{a}}_{1}, \ldots , x^{\phantom{a}}_{d} ): \text{all } x^{\phantom{a}}_{i} \in \mathbb{Z}\text{ and}\, \gcd (x^{\phantom{a}}_{1}, \ldots , x^{\phantom{a}}_{d}) = 1 \bigr\} ,\] where one enforces \(d\geqslant 2\) to avoid trivial situations. The set \(V\) has holes of arbitrary size, as follows from an application of the Chinese remainder theorem (CRT). Consequently, though being uniformly discrete, \(V\) is not relatively dense, and thus not a Delone set; see [7] for background material. In fact, since the holes appear lattice-periodically, no addition of a zero-density set can turn \(V\) into a Delone set.

The translation of \(V\) by \(t = (t^{\phantom{a}}_{1}, \ldots , t^{\phantom{a}}_{d})\in\mathbb{Z}^d\) is defined via \(t + V = \{ t + x : x \in V \}\), and the translation orbit closure \[\mathbb{X}^{\phantom{a}}_{V} \, \mathrel{\mathop{:}}=\, \overline{\mathbb{Z}^d + V}\] is a compact space, where the closure is taken in the local topology. In this special case of a Fell–Chabauty topology, two point sets in \(\mathbb{Z}^d\) are \(\varepsilon\)-close if they agree in a ball of radius \(1/\varepsilon\) around \(0\) (one can equally well use cubes, which defines the same topology). Then, the translation action of \(\mathbb{Z}^d\) on \(\mathbb{X}^{\phantom{a}}_{V}\) is continuous, and \((\mathbb{X}^{\phantom{a}}_{V}, \mathbb{Z}^d)\) is a TDS. It is far from being minimal, and \(\mathbb{X}^{\phantom{a}}_{V}\) contains the empty set, which forms the only minimal subsystem. Furthermore, \(\mathbb{X}^{\phantom{a}}_{V}\) is subset closed or hereditary, which is to say that, for each \(S\in \mathbb{X}^{\phantom{a}}_{V}\), every subset of \(S\) also lies in \(\mathbb{X}^{\phantom{a}}_{V}\); see [1], [8] and references therein for details.

Upon identifying the point set \(V\) with its characteristic function, \(\boldsymbol{1}^{\phantom{a}}_{V}\), one obtains an equivalent representation of \(\mathbb{X}^{\phantom{a}}_{V}\) as a subshift of the full shift, \(\{ 0,1\}^{\mathbb{Z}^d}\), where the natural product topology corresponds to the local topology mentioned above. We shall always identify these two pictures, thus profiting both from the geometric setting of sets and the symbolic setting of subshifts. Since \(V = \mathbb{Z}^d \setminus \bigcup_{p\in\mathcal{P}} p \mathbb{Z}^d\), we see that \(V\) consists of all points from \(\mathbb{Z}^d\) with the property that the zero coset modulo \(p \mathbb{Z}^d\) is missing, for every \(p\in\mathcal{P}\). Consequently, \(\mathbb{X}^{\phantom{a}}_{V}\) must be contained in the subshift \(\mathbb{A}\) of admissible sets, which are the subsets of \(\mathbb{Z}^d\) with the property that, for every \(p\in\mathcal{P}\), at least one coset modulo \(p \mathbb{Z}^d\) is missing. In fact, we even have \(\mathbb{X}^{\phantom{a}}_{V} = \mathbb{A}\), compare [1], [9], which is a strong property. Such subshifts are called admissible, and all examples discussed below belong to this class.

The (natural) density of \(V\) as a point set in \(\mathbb{R}^d\), denoted as \(\mathrm{dens}(V)\), is \(1/\zeta (d)\), where \(\zeta\) is Riemann’s zeta function. Here, the term ‘natural’ refers to a sequence of centred, closed balls \(B_r (0)\) with growing radius being used as averaging sequence, so \[\mathrm{dens}(V) \, = \lim_{r\to\infty} \frac{\mathrm{card}\{ x\in V : \| x \| \leqslant r \} }{\mathrm{vol}(B_r (0))} \, = \, \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{\zeta (d)}}\] holds for any \(d\geqslant 2\); compare [10] and references therein. One interesting feature is that the set \(V\) is an element of \(\mathbb{X}^{\phantom{a}}_{V}\) with maximal density. In fact, \(V\) is also maximal in the sense that no single point can be added to \(V\) without kicking it out of \(\mathbb{X}^{\phantom{a}}_{V}\). Note that one can also use centred cubes or other centred van Hove sequences, see the discussion on tied density in [10], but the density cannot be uniform, due to the existence of holes of arbitrary size in \(V \!\).

With respect to the balls of growing radius (or to any tied averaging sequence that is van Hove or Følner), also the relative patch frequencies of \(V\) exist. Here, a patch \(P\) is the intersection of \(V\) with a compact set \(B\), so \(P = V \cap B\), and its (natural) frequency emerges as the limit of counting the number of occurrences of \(P\) (up to translations) within a large ball \(B_r (0)\), divided by the volume of the ball, in the limit as \(r\to\infty\). Using these frequencies as the measures of the cylinder sets defined by the patches, they induce the (natural) patch frequency or Mirsky measure, denoted by \(\mu_{_\mathrm{M}}\), which is an invariant probability measure on \(\mathbb{X}_V\). This measure is ergodic, and \(V\) is generic for it [5], [6]. Then, \((\mathbb{X}^{\phantom{a}}_{V}, \mathbb{Z}^{d}, \mu_{_\mathrm{M}})\) is a measure-theoretic dynamical system (MTDS) with the following interesting spectral property [1], [10], where \(\mathbb{T}^d = \mathbb{R}^d /\mathbb{Z}^d = [0,1)^d\) denotes the \(d\)-torus with coordinate-wise addition modulo \(1\).

Here, the denominator of a vector \(k\in\mathbb{Q}^d\), denoted by \(\mathrm{den}(k)\), is uniquely specified by writing \(k = (k^{\phantom{a}}_{1}, \ldots , k^{\phantom{a}}_{d})/q\) with all \(k_i \in \mathbb{Z}\) and \(q\in\mathbb{N}\) subject to \(\gcd(k^{\phantom{a}}_{1}, \ldots, k^{\phantom{a}}_{d}, q)=1\), which gives \(\mathrm{den}(k) = q\). In particular, one has \(\mathrm{den}(0) = 1\). The statement on the eigenfunctions means that the system has trivial topological point spectrum, though it is known that all eigenfunctions are continuous on a subset of \(\mathbb{X}^{\phantom{a}}_{V}\) of full measure [5], [6].

For some purposes, it is advantageous to also consider \(V\) under the translation action of \(\mathbb{R}^d\). Then, the hull is \(\mathbb{Y}^{\phantom{a}}_{V} = \overline{ \{ t + V : t \in\mathbb{R}^d \} }\), again in the local topology. It is now modified by saying that two discrete point sets in \(\mathbb{R}^d\) are \(\varepsilon\)-close if they agree in \(B_{1/\varepsilon} (0)\), possibly after shifting one of them by a vector of length at most \(\varepsilon\). On the symbolic side, this corresponds to a standard suspension of \(\mathbb{Z}^d\) into \(\mathbb{R}^d\); compare [11]. The Mirsky measure extends accordingly, with cylinder sets being defined by patches up to translations in a small \(\varepsilon\)-ball, whose volume then appears as a factor to the patch frequency. Now, Theorem 1 has the following obvious counterpart; compare [1], [7], [10], [12].

This extension is also natural when considering \(V\) as a weak model set of maximal density in the sense of [5], [13]; see [2], [3] for origin and background. This approach is possible because \(x\in \mathbb{Z}^d\) is an element of \(V\) if and only if, for every rational prime \(p\), the reduction of \(x\) modulo \(p\) is not the zero element of \(\mathbb{Z}^d / p \mathbb{Z}^d\). It is now natural to consider the compact Abelian group \(H = \bigotimes_{p\in\mathcal{P}} \mathbb{Z}^d / p \mathbb{Z}^d\), where \(\mathcal{P}\) denotes the set of rational primes, and define the mapping \(\star : \mathbb{Z}^d \longrightarrow H\) by \(x \mapsto x^{\star} = \bigl( x \bmod p \mathbb{Z}^d \bigr)_{p\in\mathcal{P}}\). Then, we can characterise \(V\) as \[V \, = \, \{ x \in \mathbb{Z}^d : \text{no coordinate of } x^{\star} \text{ is } 0 \} \, = \, \{ x \in \mathbb{Z}^d : x^{\star} \in W \}\] with the coding set or window \(W = \mathop{\mathrm{\raisebox{-0.4ex}{\huge \times}}}_{\! p\in\mathcal{P}} \bigl( (\mathbb{Z}^d / p \mathbb{Z}^d)\setminus \{ 0 \} \bigr)\), which is a compact subset of \(H\). This situation is usually summarised in a cut and project scheme (CPS), which we briefly recall for the example at hand; see [7], [14] for further background and [15] for an alternative view via adeles. Here, we have \[\label{eq:vis-cps} \renewcommand{\arraystretch}{1.2}\begin{array}{r@{}ccccc@{}l} & \mathbb{R}^d & \xleftarrow{\,\;\;\pi\;\;\,} & \mathbb{R}^d \times H & \xrightarrow{\;\pi^{\phantom{a}}_{\mathrm{int}\;}} & H & \\ & \cup & & \cup & & \cup &\raisebox{1pt}{\text{ dense}} \\ & \mathbb{Z}^d & \xleftarrow{\; 1-1 \;} & \mathcal{L}& \xrightarrow{\; \hphantom{1-1} \;} & (\mathbb{Z}^d)^{\star} & \\ & \| & & & & \| & \\ & L & \multicolumn{3}{c}{\xrightarrow{\qquad\qquad\;\;\star \;\;\qquad\qquad}} & {L_{}}^{\star} & \\ \end{array}\renewcommand{\arraystretch}{1}\tag{1}\] where \(\pi\) and \(\pi^{\phantom{a}}_{\mathrm{int}}\) are the canonical projections to direct space \((\mathbb{R}^d)\) and internal space (\(H\)), respectively. Further, \(\mathcal{L}= \{ (x, x^{\star}) : x \in \mathbb{Z}^d \}\) is the diagonal embedding of \(\mathbb{Z}^d\) into \(\mathbb{R}^d {\times} H\). Note that \(\mathcal{L}\) still is a lattice in \(\mathbb{R}^d{\times}H\) (that is, a co-compact discrete subgroup). As \(\pi|^{\phantom{a}}_{\mathcal{L}}\) is a bijection between \(\mathcal{L}\) and its image, the \(\star\)-map is well defined. From now on, we denote this kind of CPS by the triple \((\mathbb{R}^d, H, \mathcal{L})\), variants of which will appear throughout the paper.

The set \(W\) is compact, but has empty interior (because \(0\) is missing from all components) and hence consists of boundary only. We are thus in the realm of weak model sets, and need to determine its type according to [5]. If \(H\) is equipped with its normalised Haar measure \(\nu_{_\mathrm{H}}\), so \(\nu_{_\mathrm{H}}(H) =1\), counting cosets modulo \(p\) gives \[\nu_{_\mathrm{H}}(W) \, = \prod_{p\in\mathcal{P}} \frac{p^d - 1}{p^d} \, = \prod_{p\in\mathcal{P}} \bigl( 1-p^{-d} \bigr) \, = \, \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{\zeta (d)}} ,\] which coincides with \(\mathrm{dens}(V)\), where the infinite product converges absolutely because we have \(\sum_{p\in\mathcal{P}} p^{-d} < \infty\) for \(d\geqslant 2\). Consequently, \(V\) is a weak model set of maximal density, because \(\mathcal{L}\) is a unimodular (co-volume one) lattice in \(\mathbb{R}^d{\times} H\). Based on this connection, one can prove Theorem 1 and Corollary 2 via the general diffraction formula for weak model sets of maximal density [5], [6] and the equivalence theorem for dynamical and diffraction spectra [16], [17]. In this setting, the relation between topological and measure-theoretic aspects is reasonably well understood; see [6] and references therein for details.

There are (at least) two independent ways to compute the dynamical spectrum, one via the countable supporting set of the pure-point diffraction measure of \(V\) (also known as the Fourier–Bohr spectrum) and another via the dual CPS [14] with the lattice \(\mathcal{L}^{0}\), which is the annihilator of \(\mathcal{L}\) from the original CPS, and its projection into \(\mathbb{R}^d\). The first method was used in [10], [18], and the second in [13]. Let us summarise the latter, as we derive a simpler method from it to compute the spectrum \(L^{\circledast}\) and then generalise this approach to other systems.

Recall that the dual group of \(\mathbb{R}^d {\times} H\) is \(\widehat{\mathbb{R}^d{\times} H} \simeq \widehat{\mathbb{R}^d} {\,\times\,} \widehat{H}\), where \(\widehat{\mathbb{R}^d} \simeq \mathbb{R}^d\) is self-dual and \[\widehat{H} \, = \, \bigoplus_{p\in\mathcal{P}} \mathbb{Z}^d/p \mathbb{Z}^d\] is the direct sum, where only finitely many coordinates are non-zero, by standard results from the harmonic analysis of LCAGs [4]. The resulting dual CPS is \[\label{eq:vis-cps-dual} \renewcommand{\arraystretch}{1.2}\begin{array}{r@{}ccccc@{}l} & \mathbb{R}^d & \xleftarrow{\,\;\;\pi\;\;\,} & \mathbb{R}^d \times \widehat{H} & \xrightarrow{\;\pi^{\phantom{a}}_{\mathrm{int}\;}} & \widehat{H} & \\ & \cup & & \cup & & \cup &\raisebox{1pt}{\text{ dense}} \\ & \pi (\mathcal{L}^{0}) & \xleftarrow{\; 1-1 \;} & \mathcal{L}^{0} & \xrightarrow{\; \hphantom{1-1} \;} & \pi^{\phantom{a}}_{\mathrm{int}}(\mathcal{L}^{0}) & \\ & \| & & & & \| & \\ & L^{\circledast} & \multicolumn{3}{c}{\xrightarrow{\qquad\qquad\;\;\star \;\;\qquad\qquad}} & (L^{\circledast})^{\star} & \\ \end{array}\renewcommand{\arraystretch}{1}\tag{2}\] which is a CPS of type \((\mathbb{R}^d, \widehat{H}, \mathcal{L}^{0})\), with \(\mathcal{L}^{0}\) the annihilator of \(\mathcal{L}\) from the above CPS \((\mathbb{R}^d, H, \mathcal{L})\) in 1 . Usually, the dual group \(\widehat{H}\) is defined as the group of continuous characters on \(H\) under pointwise multiplication. Here, we use an additive version, where \(k =(k_p)^{\phantom{a}}_{p\in\mathcal{P}}\in\widehat{H}\) stands for the continuous character \(\chi^{\phantom{a}}_{k}\) defined by \[y \mapsto \chi^{\phantom{a}}_{k} (y) \, \mathrel{\mathop{:}}=\prod_{p\in\mathcal{P}} \exp \Bigl( 2 \pi \mathrm{i}\, \frac{y_p k_p}{p} \Bigr) \qquad \text{for } y = (y_p)^{\phantom{a}}_{p\in\mathcal{P}} \in H .\] Note that the character is well defined, since only finitely many \(k_p\) differ from \(0\), so the product on the right-hand side is effectively a finite one. With this notation, the annihilator \(\mathcal{L}^{0}\) consists of all points \((u,k) \in \widehat{\mathbb{R}^d} \times \widehat{H} \simeq \mathbb{R}^d \times \widehat{H}\) such that \[\label{eq:annihilation} \exp \bigl( 2 \pi \mathrm{i}x u \bigr) \prod_{p\in\mathcal{P}} \exp \Bigl( 2 \pi \mathrm{i}\frac{x^{\star}_p k^{\phantom{a}}_p}{p} \Bigr) \, = \, 1\tag{3}\] holds for all \(x\in\mathbb{Z}^d\) or, equivalently, for all \((x,x^{\star}) \in \mathcal{L}\). Here, \(xu\) and analogous expressions stand for the standard inner product of two \(d\)-dimensional vectors.

By linearity and the rules for calculating modulo \(p\), it is clear that it suffices to satisfy this condition for \(x = e_i\) running through the standard integer basis of \(\mathbb{Z}^d\). Observing that \(u = (u^{\phantom{a}}_{1}, \ldots, u^{\phantom{a}}_{d})\) and \(k_p = (k^{\phantom{a}}_{1,p} , \ldots , k^{\phantom{a}}_{d,p})\), Eq. 3 implies coordinate-wise conditions, namely \[\label{eq:m-diff} u^{\phantom{a}}_i \, = \, m^{\phantom{a}}_{i} - \sum_{p\in\mathcal{P}} \frac{k^{\phantom{a}}_{i,p}}{p}\tag{4}\] with arbitrary \(m^{\phantom{a}}_{i}\in\mathbb{Z}\). In particular, \((u,k) \in \mathcal{L}^{0}\) implies \((u+t,k) \in \mathcal{L}^{0}\) for all \(t\in\mathbb{Z}^d\), as is clear from 3 . Since only finitely many \(k_p\) differ from \(0\), one can bring each of these conditions to a form with common denominator. Putting all conditions together, one can then see that they are equivalent to \[\label{eq:spec} u \in \{ q \in \mathbb{Q}^d : \mathrm{den}(q) \text{ is square-free} \} =\mathrel{\mathop{:}} L^{\circledast} ,\tag{5}\] which is the Fourier–Bohr (or FB) spectrum of \(V \!\). This also gives the annihilator of \(\mathcal{L}\) as \(\mathcal{L}^{0} = \{ ( u, u^{\star}) : u \in L^{\circledast} \}\), where the \(\star\)-map is that of the dual CPS, as defined by \[L^{\circledast} \ni u \, \longmapsto \, u^{\star} \mathrel{\mathop{:}}=\bigl( - \mathrm{lcm}( \mathrm{den}(u), p) u \bmod p \bigr)_{p\in\mathcal{P}} \, ,\] where the entry at \(p\) is \(- \mathrm{den}(u) u \bmod p\) for all \(p \mid \mathrm{den}(u)\) and \(0\) otherwise. This mapping emerges from taking all \(m^{\phantom{a}}_i =0\) in 4 without loss of generality, because \((u+t)^{\star} = u^{\star}\) for all \(t\in\mathbb{Z}^d\). More precisely, \(\star \colon L^{\circledast} \xrightarrow{\quad} \widehat{H}\) is a group homomorphism with kernel \(\mathbb{Z}^d\). Strictly speaking, we now have two \(\star\)-maps, one for the original CPS and one for its dual. Nevertheless, we shall use the same symbol for both, as misunderstandings are unlikely.

The set \(L^{\circledast}\) is also the dynamical spectrum of the \(\mathbb{R}^d\)-flow induced by \(V\), compare [17], and its calculation via the dual CPS will be possible more generally. In fact, this is where Remark 3 will become quite crucial. Likewise, the group \(L^{\circledast}\! / \mathbb{Z}^d\) from Remark 4 is the dynamical spectrum for the discrete group action of \(\mathbb{Z}^d\). Another advantage of the connection with diffraction theory and a CPS is that this setting also allows for a closed form of the eigenfunctions (relative to the Mirsky measure \(\mu_{_\mathrm{M}}\)) as follows. For \(y\in \mathbb{R}^d\), we define the corresponding FB coefficient of \(V\) as \[\label{eq:FB-def-1} a^{\phantom{a}}_{V} (y) \, \mathrel{\mathop{:}}=\lim_{r\to\infty} \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{\mathrm{vol}(B_r (0))}} \sum_{ x \in V_r } \mathrm{e}^{- 2 \pi \mathrm{i}xy}, \qquad \text{with } V_r = V \cap B_r (0) .\tag{6}\] The limit exists for all \(y\in\mathbb{R}^d\), as can be shown with an explicit convergence argument. The latter is based on approximating \(V\) by a nested sequence of periodic point sets that are obtained from taking only the first \(N\) primes into account and then letting \(N\to\infty\). This is a natural way to view the limit, as it actually shows how the coefficients emerge from those of a series of approximating periodic systems.

The volume-averaged exponential sum in 6 is nontrivial for any \(y\in L^{\circledast}\), which is a countable set, and vanishes everywhere else [10], [13]. Further, one observes the behaviour under translations as \[a^{\phantom{a}}_{t+V} (y) \, = \, \mathrm{e}^{-2 \pi \mathrm{i}t y} a^{\phantom{a}}_{V} (y) .\] So, given \(y\in L^{\circledast}\), we have an eigenvector equation along the translation orbit of \(V\!\), with eigenvalue \(\lambda_y = \mathrm{e}^{-2\pi\mathrm{i}ty}\) on the unit circle, where \(y\) provides the advantageous additive notation mentioned earlier. The corresponding definition and eigenvalue equation applies to all other elements of the hull for which the limit exists, which are most of them in a measure-theoretic sense, not only for \(\mu_{_\mathrm{M}}\), in ways we will explain later; see [19], [20] for the required notions of genericity and their relation to Besicovitch almost periodicity.

When we consider the \(\mathbb{Z}^d\)-action, it follows from the results of [5], [6] that, except for the trivial eigenfunction (which corresponds to \(y=0\)), no eigenfunction can possess a continuous representative, but that they are continuous on a subset of \(\mathbb{X}^{\phantom{a}}_{V}\) of full measure. This is related to \(\mathbb{X}^{\phantom{a}}_{V}\) being a limit of periodic systems (with pure-point spectrum and continuous eigenfunctions) in such a way that the spectrum of \((\mathbb{X}^{\phantom{a}}_{V},\mathbb{Z}^d,\mu_{_\mathrm{M}})\) is still obtained as a limit, but continuity of the eigenfunctions is lost. Under the suspension of \(\mathbb{X}^{\phantom{a}}_{V}\) to \(\mathbb{Y}^{\phantom{a}}_{V}\) for the translation action of \(\mathbb{R}^d\), precisely the eigenfunctions for \(y\in \mathbb{Z}^d\) are continuous, but none of the others. Still, it is sufficient to know them on the translation orbit of sufficiently many elements of the hull, for instance for the generic elements for \(\mu_{_\mathrm{M}}\), which is a set of full measure. Here, we derive the FB coefficients for \(V\), where we have the following result.

Two comments are in order. First, in this representation, all eigenfunctions take only real values on \(V\!\). This simply reflects the mirror symmetry of \(V\!\), which turns the FB coefficients into a volume-averaged cosine sum. Then, the normalisation of the eigenfunctions is not the standard (unimodular) one, but very natural from the harmonic analysis point of view. In fact, the diffraction measure [10] of \(V\) is the positive, pure-point measure \[\widehat{\gamma} \, = \sum_{y\in L^{\circledast}} \lvert a^{\phantom{a}}_{V} (y) \rvert^2 \, \delta^{\phantom{a}}_y ,\] which gives the intensities of the peaks as the squares (of the absolute values) of the FB coefficients, which is also known as the phase consistency of the system [19], while the supporting set of \(\widehat{\gamma}\) is the FB spectrum, which agrees with the dynamical point spectrum in our case,¹ where we do not take the closure as otherwise done in functional analysis.

Let us also take a brief look at the related MTDS \((\mathbb{Y}^{\phantom{a}}_{V},\mathbb{R}^d,\mu^{\phantom{a}}_{\max})\), where \(\mu^{\phantom{a}}_{\max}\) is the measure of maximal entropy. It reflects the hereditary nature of \(\mathbb{Y}^{\phantom{a}}_{V}\), and is unique. It comes from the product of \(\mu_{_\mathrm{M}}\) with the Bernoulli measure induced by the fair coin toss [22]–[24]. The new measure emerges from considering the product system \(\mathbb{X}^{\phantom{a}}_{V} \times \{0,1\}^{\mathbb{Z}^d}\) with \(\mu_{_\mathrm{M}}\) on the first and the Bernoulli measure on the second factor. Then, since \(\mathbb{X}^{\phantom{a}}_{V}\) is hereditary, pointwise multiplication is well defined and maps any pair \((x,z)\) to \(xz\), which is always in \(\mathbb{X}^{\phantom{a}}_{V}\).

This new MTDS has mixed spectrum, as a result of [25], [26], with a pure-point part that is closely related to the pure-point spectrum discussed above and an absolutely continuous part that corresponds to countably many copies of Lebesgue measure. Let us consider the pure-point part constructively, which is possible because \(V\) can be turned into a generic element of \(\mathbb{Y}^{\phantom{a}}_{V}\) relative to \(\mu^{\phantom{a}}_{\max}\) by a standard Bernoulli thinning. This refers to knocking out each point \(x\in V\) randomly and independently with probability \(\frac{1}{2}\), and almost every realisation of this process will lead to a generic element for \(\mu^{\phantom{a}}_{\max}\). Clearly, this Bernoulli thinning also works for any other \(\mu_{_\mathrm{M}}\)-generic element of \(\mathbb{Y}_V\), not just for \(V\!\). Consequently, the set of elements of \(\mathbb{Y}^{\phantom{a}}_{V}\) we reach this way has full measure with respect to \(\mu_{\max}\).

More generally, let \(p\in (0,1)\) and let \((\xi^{\phantom{a}}_{x})^{\phantom{a}}_{x\in V}\) be a family of i.i.d. random variables with values in \(\{ 0,1 \}\) and \(\mathbb{P}(\xi^{\phantom{a}}_{x} =1) = p\). Now, let the set \(\widetilde{V} = \{ x \in V \! : \xi^{\phantom{a}}_{x} = 1 \}\) be a realisation of the corresponding thinning, which is almost surely generic for the pushforward of the product measure of \(\mu_{_\mathrm{M}}\) with the Bernoulli measure for \(p\), where \(p=\frac{1}{2}\) corresponds to \(\mu^{\phantom{a}}_{\max}\). What is more, such a realisation almost surely has (natural) density \(\mathrm{dens}\bigl(\widetilde{V}\bigr) =p\, \mathrm{dens}(V)\), both claims following by an application of the strong law of large numbers (SLLN); see the Appendix for further details. It is now possible to determine the FB coefficients of \(\widetilde{V}\!\), where we get \[\label{eq:FB-def} a_{\widetilde{V}} (y) \, = \lim_{r\to\infty} \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{\mathrm{vol}( B_{r} (0) )}} \sum_{x\in \widetilde{V}_{r}} \mathrm{e}^{-2\pi\mathrm{i}xy} \, = \lim_{r\to\infty} \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{\mathrm{vol}( B_{r} (0) )}} \sum_{x\in V_{r}} \xi^{\phantom{a}}_{x} \, \mathrm{e}^{-2\pi\mathrm{i}xy} .\tag{7}\]

The complex random variables \(Z^{\phantom{a}}_{x} \mathrel{\mathop{:}}=\xi^{\phantom{a}}_{x} \mathrm{e}^{-2\pi\mathrm{i}xy}\) are independent, but no longer identically distributed. With values in \(\{0 \} \cup \mathbb{S}^{1}\), they clearly have bounded variance. Indeed, one has \(\mathbb{E}(Z^{\phantom{a}}_x) = \mathbb{E}(\xi^{\phantom{a}}_{x}) \mathrm{e}^{-2\pi\mathrm{i}xy} = p \, \mathrm{e}^{-2\pi\mathrm{i}xy}\) and \[\mathbb{V} (Z^{\phantom{a}}_{x}) \, = \, \mathbb{E}\bigl( \lvert Z_x - \mathbb{E}(Z_x) \rvert^2 \bigr) \, = \, \mathbb{E}\bigl( \lvert Z_x \rvert^2\bigr) - \big\lvert\mathbb{E}(Z_x) \big\rvert^2 \, = \, p (1-p) .\] Consequently, we can apply Kolmogorov’s version of the SLLN, compare [27], so we may replace each term \(\xi^{\phantom{a}}_{x} \mathrm{e}^{-2\pi\mathrm{i}xy}\) by its expectation, and almost surely still get the same limit. For any fixed \(y\in\mathbb{R}^d\), this almost surely gives \[\label{eq:thinned-functions} a_{\widetilde{V}} (y) \, = \, p \cdot a^{\phantom{a}}_{V} (y) .\tag{8}\] Since \(V\) can be replaced by any \(\mu_{_\mathrm{M}}\)-generic element, this procedure works for a subset of the hull that has full measure for \(\mu^{\phantom{a}}_{\max}\). As explained earlier, a non-zero \(a_{\widetilde{V}} (y)\) induces an eigenfunction, which can happen for at most countably many \(y\in\mathbb{R}^d\). We thus see that, almost surely, Eq. 8 holds for all \(y\in L^{\circledast}\) together with \(a_{\widetilde{V}} (y) =0\) for all other \(y\). This is to say that the point spectrum of \((\mathbb{Y}^{\phantom{a}}_{V}, \mathbb{R}^d, \mu_{_\mathrm{M}})\) agrees with that of \((\mathbb{Y}^{\phantom{a}}_{V}, \mathbb{R}^d, \mu^{\phantom{a}}_{\max})\), with the eigenfunctions emerging from the FB coefficients in both cases (the prefactor \(p\) only changes the normalisation). The difference is that these eigenfunctions now only span a subspace of \(L^2 (\mathbb{Y}^{\phantom{a}}_{V}, \mu^{\phantom{a}}_{\max})\). Its complement corresponds to the absolutely continuous part of the spectrum, with countably many copies of Lebesgue measure as spectral measures. The above considerations can be summarised as follows.

To go one step further, we can employ the Bernoullisation method from [7] to also calculate the almost sure diffraction measure of \(( \mathbb{Y}^{\phantom{a}}_{V}, \mu^{\phantom{a}}_{\max})\), which gives \[\widehat{\gamma}^{\phantom{a}}_{\max} \, = \, \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{4}} \widehat{\gamma} + \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{4}} \mathrm{dens}(V) \lambda_{_\mathrm{L}} ,\] where \(\lambda_{_\mathrm{L}}\) denotes Lebesgue measure. This, together with 8 shows constructively that the new intensities of the peaks in the pure-point part are given by the absolute value squares of the new FB coefficients. Thus, the pure-point part still satisfies phase consistency.

Two comments are in order here. First, it is clear why working with the Mirsky measure is both natural and sufficient, which is what we will do throughout the paper. Second, in our special situation, there is a deeper picture behind this, which revolves around uniformly distributed sequences. This will be explained in more detail in the Appendix.

3 Power-free integers and some of their generalisations↩︎

The simplest one-dimensional analogue of the visible lattice points is the set of square-free integers. This set contains all integers that are not divisible by the square of any (rational) prime, and thus constitutes a special case of the class of \(\mathcal{B}\)-free integers, with \(\mathcal{B}= \{ p^2 : p \in \mathcal{P}\}\). Square-free integers have been studied for a long time, see [10] and references therein for early results and [7], [13], [22] for more recent treatments, while we refer to [9], [28]–[31] and references therein for recent results on \(\mathcal{B}\)-free systems, and [32], [33] for some results on the symmetries.

A simple extension is given by \[V_{\boldsymbol{\kappa}} \, \mathrel{\mathop{:}}=\, \{ x \in \mathbb{Z}: x \text{ not divisible by } p^{\kappa_p} \text{ for any } p \in \mathcal{P}\} ,\] where \(\boldsymbol{\kappa} = (\kappa_p)^{\phantom{a}}_{p\in\mathcal{P}}\) is a sequence of natural numbers with some constraints. In particular, \(\kappa_p =1\) is only allowed for finitely many \(p\in\mathcal{P}\), or in such a way that \[\label{eq:sum-restriction} \sum_{p \, : \kappa_p =1}p^{-1} \, < \, \infty .\tag{9}\] Further, we can also admit \(\kappa_p = \infty\), which just means that those primes are not in \(\mathcal{B}\) and thus give no condition. When only finitely many \(\kappa_p\) are finite, \(V_{\boldsymbol{\kappa}}\) is a periodic subset of \(\mathbb{Z}\). In general, when 9 holds, the natural density (via averaging over intervals \([-n,n]\) as \(n\to\infty\)) of \(V_{\boldsymbol{\kappa}}\) exists and is given by the generally infinite, but absolutely convergent, product \[\mathrm{dens}(V_{\boldsymbol{\kappa}}) \, = \prod_{p\in\mathcal{P}} \bigl( 1-p^{-\kappa_p} \bigr) \, = \prod_{p\in\mathcal{P}'} \bigl( 1 - p^{-\kappa_p}\bigr) ,\] with \(\mathcal{P}' \mathrel{\mathop{:}}=\{ p \in \mathcal{P}: \kappa_p < \infty \}\), as follows from a simple inclusion-exclusion argument via the periodic covering sets that are obtained from taking the first \(N\) primes into account and then letting \(N\to\infty\), in conjunction with an appropriate tail estimate; see Section 4 below. When \(\kappa_p = \kappa\) with \(\kappa \geqslant 2\) for all \(p\in\mathcal{P}\), we simply write \(V_{\kappa}\), and one gets \(\mathrm{dens}(V_{\kappa}) = 1/\zeta (\kappa)\).

For the general case, a CPS for \(V_{\boldsymbol{\kappa}}\) can be constructed as \[\label{eq:int-cps} \renewcommand{\arraystretch}{1.2}\begin{array}{r@{}ccccc@{}l} & \mathbb{R}& \xleftarrow{\,\;\;\pi\;\;\,} & \mathbb{R}\times H & \xrightarrow{\;\pi^{\phantom{a}}_{\mathrm{int}\;}} & H & \\ & \cup & & \cup & & \cup &\raisebox{1pt}{\text{ dense}} \\ & \mathbb{Z}& \xleftarrow{\; 1-1 \;} & \mathcal{L}& \xrightarrow{\; \hphantom{1-1} \;} & \iota (\mathbb{Z}) & \\ & \| & & & & \| & \\ & L & \multicolumn{3}{c}{\xrightarrow{\qquad\qquad\;\,\star \,\;\qquad\qquad}} & {L_{}}^{\star} & \\ \end{array}\renewcommand{\arraystretch}{1}\tag{10}\] where \(H=\bigotimes_{p\in\mathcal{P}'} \mathbb{Z}/p^{\kappa_p} \mathbb{Z}\) is a compact Abelian group and \(\pi\) and \(\pi^{\phantom{a}}_{\mathrm{int}}\) are the canonical projections to \(\mathbb{R}\) and \(H\), respectively. Further, \(\mathcal{L}\) is a lattice in \(\mathbb{R}{\times} H\), where \(\iota : \mathbb{Z}\longrightarrow H\) is the mapping defined by \(n \mapsto n^{\star} \mathrel{\mathop{:}}=( n \bmod p^{\kappa_p} )^{\phantom{a}}_{p\in \mathcal{P}'}\) and \(\mathcal{L}\) is the diagonal embedding of \(\mathbb{Z}\), so \(\mathcal{L}= \{ (n, n^{\star}) : n \in \mathbb{Z}\}\). In other words, this CPS is a simple variant of 1 , with the adjusted triple \((\mathbb{R}, H, \mathcal{L})\). Then, we get \[V_{\boldsymbol{\kappa}} \, = \, \{ n \in \mathbb{Z}: n^{\star} \in W \} \qquad \text{with } \; W \, = \mathop{\mathrm{\raisebox{-0.7ex}{\Huge \times}}}\displaylimits_{p\in \mathcal{P}'} \bigl( ( \mathbb{Z}/p^{\kappa_p} \mathbb{Z}) \setminus \{ 0 \} \bigr) .\] If \(H\) is equipped with its normalised Haar measure \(\nu_{_\mathrm{H}}\), the window \(W\) has measure \[\nu_{_\mathrm{H}}(W) \, = \prod_{p\in\mathcal{P}'} \frac{p^{\kappa_p} - 1}{p^{\kappa_p}} \, = \, \mathrm{dens}(V_{\boldsymbol{\kappa}}) ,\] which shows the analogy to our guiding example.

By a minor extension of the arguments used in [10] for the case of constant \(\kappa_p\), one can derive the diffraction measure of the Dirac comb \(\delta^{\phantom{a}}_{V_{\boldsymbol{\kappa}}} = \sum_{x\in V_{\boldsymbol{\kappa}}} \delta_x\) as \[\widehat{\gamma^{\phantom{a}}_{V_{\boldsymbol{\kappa}}}} \, = \sum_{k\in L^{\circledast}} \lvert a(k) \rvert^{2} \delta^{\phantom{a}}_{k}\] with the FB coefficients \(a(k)\). The supporting set, the FB spectrum, is \[\label{eq:kappa-spec} L^{\circledast} \, = \, \Bigl\{ k \in \mathbb{Q}: \begin{array}{c} \mathrm{den}(k) \text{ is (\kappa_p {+} 1)-free for all } p \in \mathcal{P}' \\ \text{and not divisible by any } q \in \mathcal{P}\setminus\mathcal{P}' \end{array} \Bigr\} \, = \, \sum_{p\in\mathcal{P}'} \, p^{-\kappa_p} \mathbb{Z},\tag{11}\] which is a subgroup of \(\mathbb{Q}\), hence also torsion-free. It is again calculated by the method from Remark 3. Further, analogously to Theorem 5, the FB coefficients for \(k\in L^{\circledast}\) are given by \[\label{eq:FB-kappa-free} a(k) \, = \mathrm{dens}(V_{\boldsymbol{\kappa}}) \prod_{p \mid \mathrm{den}(k)} \frac{1}{1-p^{\kappa_p}} ,\tag{12}\] while they satisfy \(a (k)=0\) for all other \(k\). Note that \(a (k)\) is always real, which is clear from the reflection symmetry \(V_{\boldsymbol{\kappa}} = - V_{\boldsymbol{\kappa}}\). The group \(L^{\circledast}\) is also the dynamical spectrum (in additive notation) of the MTDS \((\mathbb{Y}_{\boldsymbol{\kappa}}, \mathbb{R}, \mu_{_\mathrm{M}})\) with \(\mathbb{Y}_{\boldsymbol{\kappa}}\) being the continuous hull of \(V_{\boldsymbol{\kappa}}\) and \(\mu_{_\mathrm{M}}\) the Mirsky measure. Restricting to \(\mathbb{Z}\)-action, one gets the spectrum \(L^{\circledast}\cap \mathbb{T}\) with \(\mathbb{T}\) as above, in complete analogy to Remark 4. Under addition modulo \(1\), this is the group \[L^{\circledast}\! / \mathbb{Z}\, = \, \bigoplus_{p\in \mathcal{P}'} p^{-\kappa_{p}} \mathbb{Z}/ \mathbb{Z}, \quad\text{with}\quad p^{-\kappa_{p}} \mathbb{Z}/ \mathbb{Z}\simeq C_{p^{\kappa_p}} .\] Conversely, when starting from this discrete group action, the extension to the \(\mathbb{R}\)-action emerges via a standard suspension with a constant roof function; compare [11].

Again, the dynamical spectrum can also be extracted from the dual CPS, which is a modification of 2 , now with \(d=1\) and adjusted entries for \(H\) and $^0 \(. The latter is\)\(\mathcal{L}^{0} \, = \, \{ (y,y^{\star}) : y \in L^{\circledast} \}\)$ with \(\star \colon L^{\circledast} \xrightarrow{\quad} \widehat{H} = \bigoplus_{p\in\mathcal{P}'} \mathbb{Z}/p^{\kappa_{p}}\mathbb{Z}\) being defined as follows. First, consider the terms in \(\prod_{p \mid \mathrm{den}(u)} p^{\kappa_p}\) and then set \[u^{\star} \, = \, \bigl( - \mathrm{lcm}(\mathrm{den}(u), p^{\kappa_p}) u \bmod p^{\kappa_p} \bigr)_{p\in\mathcal{P}'} ,\] which is a slight modification of our \(\star\)-map for the visible lattice points. Once again, the entry for any \(p \nmid \mathrm{den}(u)\) vanishes, so that \(\mathbb{Z}\) is the kernel of the group homomorphism \(\star\). All of this follows from calculations analogous to those of our guiding example, which we leave to the interested reader; see also [13]. Note that this formulation is also correct when \(\mathcal{P}'\) is a finite set, in which case \(V_{\boldsymbol{\kappa}}\) is periodic. The resulting expressions can then also be seen via a simple inclusion-exclusion argument, which is the basis of the argument from Remark 3.

Let us summarise the above derivations and recollections as follows.

At this point, our previous comments on \(\mu_{_\mathrm{M}}\) versus \(\mu^{\phantom{a}}_{\max}\) apply in complete analogy. In particular, we could repeat Corollary 7 for this systems as well, which we omit.

4 A more general setting with \(\mathcal{B}\)-free lattice systems↩︎

Though we are particularly interested in the number-theoretic setting, a more general approach would employ lattices. Indeed, consider \(\mathbb{R}^d\), with some fixed norm \(\| . \|\), and fix some lattice \(\varGamma\) in it. The choice \(\varGamma= \mathbb{Z}^d\) will then correspond to the situation of symbolic dynamics. Now, select a collection of lattices \(\mathcal{B}= \{ \varGamma_i : i \in \mathbb{N}\}\) with the following properties.

• One has \(\varGamma_i \subset \varGamma\) with \(1 < [\varGamma: \varGamma_i] \leqslant [\varGamma: \varGamma_{i+1}]\) for all \(i\in \mathbb{N}\).

• One has \((\varGamma_1 \cap \ldots \cap \varGamma_m) + \varGamma_{m+1} = \varGamma\) for all \(m \in \mathbb{N}\).

• One has \(\sum_{i\in\mathbb{N}} \lambda (\varGamma_i)^{-d} < \infty\), where \(\lambda (\varGamma) = \min_{x\in \varGamma\setminus \{ 0 \}} \| x \|\).

Note that (B2) implies \(\varGamma_i + \varGamma_j = \varGamma\) for all \(i \ne j\). The latter is equivalent to (B2) when \(d=1\), but not for \(d>1\). There, (B2) is a stronger condition, which is equivalent to pairwise coprimeness in conjunction with the condition that the natural projection of \(\varGamma\) into \(H = \bigotimes_{i\in\mathbb{N}} \varGamma/ \varGamma_i\) is dense. The latter is true if and only if \(\varGamma\) projects surjectively to \(\bigotimes_{i=1}^{m} \varGamma/\varGamma_i\) for all \(m\in \mathbb{N}\). In fact, (B2) ensures the lattice version of the CRT, which we need in our density arguments.

Now, we consider \(V = \varGamma\setminus \bigcup_{i\in\mathbb{N}} \varGamma_i\), which generalises our previous examples. Also all later ones will be of this type. These conditions are one possible generalisation of Erdős \(\mathcal{B}\)-free systems [24], [28] from one to higher dimensions [1]. We note that (B3) is sufficient for what we need, but not necessary for \(d>1\); slightly weaker conditions will still work, but are not pursued here. Let us mention here that the above setting can be extended to cover more general sieves. For instance, one could be interested in removing not just \(\varGamma_i\), but also some cosets of it in \(\varGamma\), which we leave for future work.

Set \(V_n = \varGamma\setminus \bigcup_{i=1}^{n} \varGamma_i\), which is still a periodic point set, with \(\bigcap_{i=1}^{n} \varGamma_i\) as its lattice of periods. Note that \(V^{\phantom{a}}_{0} \mathrel{\mathop{:}}=\varGamma\) is a limiting case, with \(\varGamma\) as its lattice of periods. The density of any \(V_n\) exists uniformly. With \([n]\mathrel{\mathop{:}}=\{ 1, \ldots , n \}\) for \(n\in\mathbb{N}\) and \([0]\mathrel{\mathop{:}}=\varnothing\), one has \[\mathrm{dens}(V_n) \, = \sum_{I\subseteq [n]} (-1)^{\lvert I \rvert} \mathrm{dens}(\varGamma^{\phantom{a}}_I)\] by a standard inclusion-exclusion argument, where \(\varGamma^{\phantom{a}}_I = \bigcap_{i\in I} \varGamma_i\) with \(\varGamma^{\phantom{a}}_{\varnothing} \mathrel{\mathop{:}}=\varGamma\) and \(\lvert I \rvert\) denotes the cardinality of the finite set \(I\). Due to condition (B2), we simply have \[\mathrm{dens}(\varGamma^{\phantom{a}}_{I}) \, = \, \mathrm{dens}(\varGamma) \prod_{i\in I} [\varGamma: \varGamma_i]^{-1} ,\] which then gives, with \(N_i \mathrel{\mathop{:}}=[\varGamma: \varGamma_i]\), \[\label{eq:dens-Vn} \mathrm{dens}(V_n) \, = \mathrm{dens}(\varGamma) \prod_{i=1}^{n} \Bigl( 1 - \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{N_{i}}} \Bigr) .\tag{13}\] As \(n\to\infty\), the product converges as a consequence of (B3), because \(N_i \geqslant \lambda (\varGamma_{i})^d\).

In contrast, the density of \(V\) will generally not be uniform, and is more complicated. Since \(V\subset V_n\) for every \(n\in \mathbb{N}\), we get \(\overline{\mathrm{dens}} (V) \leqslant \mathrm{dens}(V_n)\) for all \(n\in\mathbb{N}\), where the upper density is defined with respect to any given Følner averaging sequence. This implies \[\overline{\mathrm{dens}} (V) \, \leqslant \, \inf_{n\in\mathbb{N}} \mathrm{dens}(V_n) \, = \lim_{n\to\infty} \mathrm{dens}(V_n) \, = \, \mathrm{dens}(\varGamma) \prod_{i=1}^{\infty} \Bigl( 1 - \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{N_i }} \Bigr) ,\] where the first equality follows from \(V_n \supset V_{n+1}\) for all \(n\in\mathbb{N}\), and the second from 13 .

In general, the corresponding lower density could differ, but our special conditions are chosen so that this does not happen for the natural (tied) density with respect to centred balls. So, let \(B_r = B_r (0)\) be the closed ball of radius \(r\) around \(0\) and consider \(\mathcal{A}= \{ B_r : r > 0 \}\). For any point set \(S\subset \mathbb{R}^d\), we now use \[\overline{\mathrm{dens}}_{\mathcal{A}} (S) \, \mathrel{\mathop{:}}=\limsup_{r\to\infty} \frac{\raisebox{-2pt}{\lvert S\cap B_r \rvert}}{\raisebox{0.5pt}{\mathrm{vol}(B_r)}} \quad \text{and} \quad \underline{\mathrm{dens}}_{\mathcal{A}} (S) \, \mathrel{\mathop{:}}= \liminf_{r\to\infty} \frac{\raisebox{-2pt}{\lvert S\cap B_r \rvert}}{\raisebox{0.5pt}{\mathrm{vol}(B_r)}}\] and say that the density of \(S\) with respect to \(\mathcal{A}\) exists if \(\overline{\mathrm{dens}}_{\mathcal{A}} (S) = \underline{\mathrm{dens}}_{\mathcal{A}} (S)\). From the above, we get \[\overline{\mathrm{dens}}_{\mathcal{A}} (V) \, \leqslant \, \overline{\mathrm{dens}}_{\mathcal{A}} (V_n) \, = \, \mathrm{dens}_{\mathcal{A}} (V_n) \, = \, \mathrm{dens}(V_n)\] for every \(n\in\mathbb{N}\), where \(\mathrm{dens}(V_n)\) refers to the uniform density of \(V_n\) from above. Since \(V\subset V_n\), we have the disjoint union \(V_n = V \dot{\cup}\varDelta_n\) with \(\varDelta_n = V_n \!\setminusV\).

Now, let \((A_i)^{\phantom{a}}_{i\in\mathbb{N}}\) with \(A_i = B_{r_i}\) be a sequence of growing balls (with unbounded radii) such that \(\underline{\mathrm{dens}}_{\mathcal{A}} (V) = \lim_{i\to\infty} \frac{\lvert V \cap A_i \rvert}{\mathrm{vol}(A_i)}\), which must exist by standard arguments. Observing that \(V_n \cap A_i = (V \cap A_i) \, \dot{\cup} \, (\varDelta_n \cap A_i)\), we get the estimate \[\mathrm{dens}(V_n) \, = \lim_{i\to\infty} \frac{\raisebox{-2pt}{\lvert V_n \cap A_i \rvert}}{\raisebox{0.5pt}{\mathrm{vol}(A_i)}} \, \leqslant \, \underline{\mathrm{dens}}_{\mathcal{A}} (V) + \overline{\mathrm{dens}}_{\mathcal{A}} (\varDelta_n)\] and hence \(\underline{\mathrm{dens}}_{\mathcal{A}} (V) \geqslant \mathrm{dens}(V_n) - \overline{\mathrm{dens}}_{\mathcal{A}} (\varDelta_n)\) for all \(n\in\mathbb{N}\). Now, the upper density of \(\varDelta_n\) gets arbitrarily small with growing \(n\), as a consequence of condition (B3), so lower and upper density of \(V\) with respect to \(\mathcal{A}\) must agree, and we have \[\mathrm{dens}_{\mathcal{A}} (V) \, = \lim_{n\to\infty} \mathrm{dens}(V_n) .\] Invoking the results from [5], this has the following consequence.

Note that each \(\varGamma^{*}_{i}/\varGamma^{*}\) is a finite Abelian group, though its decomposition into cyclic groups is not possible without further information on the lattices in \(\mathcal{B}\).

The corresponding TDS in the setting of symbolic dynamics is obtained via the orbit closure \(\mathbb{X}^{\phantom{a}}_V\) of \(1^{\phantom{a}}_{V}\) under the translation action of \(\varGamma\), which gives \((\mathbb{X}^{\phantom{a}}_V, \varGamma)\). The space \(\mathbb{X}^{\phantom{a}}_{V} \subset \{ 0, 1\}^{\varGamma}\) is compact and known as the discrete hull of \(V\) under the action of \(\varGamma\). Its flow counterpart is \((\mathbb{Y}^{\phantom{a}}_{V}, \mathbb{R}^d)\), with \(\mathbb{Y}^{\phantom{a}}_{V}\) the orbit closure under the translation action of \(\mathbb{R}^d\) as the continuous hull. When equipped with the Mirsky measure, which is once again obtained as the patch frequency measure derived from \(V\) relative to a tied Følner averaging sequence \(\mathcal{A}\), we get pure-point dynamical spectrum. It is, in additive notation, given by \(L^{\circledast}\cap \mathbb{T}^d\) for \((\mathbb{X}^{\phantom{a}}_{V},\varGamma, \mu_{_\mathrm{M}})\) and by \(L^{\circledast}\) for \((\mathbb{Y}^{\phantom{a}}_{V}, \mathbb{R}^d, \mu_{_\mathrm{M}})\), respectively. No eigenfunction of \((\mathbb{X}^{\phantom{a}}_{V}, \varGamma, \mu_{_\mathrm{M}})\), except the trivial one, admits a continuous representation. For \((\mathbb{Y}^{\phantom{a}}_{V}, \mathbb{R}^d, \mu_{_\mathrm{M}})\), the only continuous eigenfunctions are the ones for the eigenvalues from \(\varGamma^{*}\!\). Still, starting from \(V\), which is a generic set for both dynamical systems, the eigenfunctions can be given by the FB coefficients.

To compute them, we first recall that, for any finite \(I\subset \mathbb{N}\), the FB coefficient of \(\varGamma^{\phantom{a}}_{I}\) is \[a^{\phantom{a}}_{\varGamma^{\phantom{a}}_{\! I}} (k) \, = \lim_{r\to\infty} \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{\mathrm{vol}(B_r)}} \sum_{x\in\varGamma^{\phantom{a}}_{\! I}\cap B_r} \! \mathrm{e}^{-2 \pi \mathrm{i}kx} \, = \, \begin{cases} \mathrm{dens}(\varGamma^{\phantom{a}}_{I}) , & \text{if } k\in\varGamma^{*}_{I}, \\ 0 , & \text{otherwise}. \end{cases}\] This gives the FB coefficients of \(V_n\) via inclusion-exclusion as \(a^{\phantom{a}}_{V_n} (k) = \sum_{I \subseteq [n]} (-1)^{\lvert I \rvert} a^{\phantom{a}}_{\varGamma^{\phantom{a}}_{\! I}} (k)\), which vanishes for all \(k \notin \varGamma^{*}_{[n]}\). This formula also holds in the limit as \(n\to\infty\), where one gets non-zero results only for \(k\in L^{\circledast}\). If \(k\in L^{\circledast}\), one can define \[D (k) \, = \, \{ i\in\mathbb{N}: k \in \varGamma^{*}_{i} \setminus \varGamma^{*} \} ,\] which is a finite set and takes the role of the denominator of a rational vector in our guiding examples, due to the property (B2). Then, for all \(k\in L^{\circledast}\), one finds \[\label{eq:gen-FB} a^{\phantom{a}}_{V} (k) \, = \, \mathrm{dens}^{\phantom{a}}_{\mathcal{A}} (V) \prod_{i\in D(k)} \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{1 - N_{i} }}\tag{14}\] together with \(a^{\phantom{a}}_{V} (k) = 0\) for all other \(k\). The verification of this formula uses the very same inclusion-exclusion calculation that underlies the FB coefficient in Theorem 5.

The eigenfunction property follows once again from the relation \(a^{\phantom{a}}_{t+V} (k) = \mathrm{e}^{-2 \pi \mathrm{i}k t} a^{\phantom{a}}_{V} (k)\) in conjunction with the fact that \(V\) is a generic element for both hulls. Putting everything together, we have the following result.

While these results look nice and decently general, it is not possible to compute the spectrum much further unless additional properties are present. Fortunately, this is the case in the number-theoretic setting, as we shall now discuss in detail for one class of examples.

5 Quadratic number fields↩︎

Quadratic number fields come in two flavours, real and imaginary, which require slightly different computations. After some recollections with focus on our needs, we will thus treat them separately; see [34] for general background.

Any quadratic number field is of the form \(K = \mathbb{Q}(\sqrt{d}\,)\) for some integer \(1 \ne d \in \mathbb{Z}\) that is square-free. The ring of integers in \(K\), which is a maximal order, is \[\label{eq:order} \mathcal{O}^{\phantom{a}}_{K} \, = \, \begin{cases} \mathbb{Z}\bigl[ \frac{1 + \sqrt{d}}{2} \bigr], & \text{if d \equiv 1 \bmod 4}, \\ \mathbb{Z}[ \sqrt{d}\,], & \text{otherwise}. \end{cases}\tag{15}\] When \(d<0\), the unit group \(\mathcal{O}^{\times}_{K}\) is finite, and isomorphic to \(C_4\) (\(d=-1\)), to \(C_6\) (\(d=-3\)), or to \(C_2 = \{ \pm 1 \}\) (in all remaining cases). In contrast, \(\mathcal{O}^{\times}_{K} \simeq C_2 \times C_{\infty}\) for all real quadratic fields, where the \(C_{\infty}\) is generated by a fundamental unit.

Let us recall a useful result that gives access to the Abelian factor groups \(\mathcal{O}/\mathfrak{p}^{\kappa}\) for a prime ideal \(\mathfrak{p}\) and \(\kappa\in\mathbb{N}\), which follows from the material in [35].

Let us next state a result that will allow us to verify property (B3) in our later examples. From now on, we shall mostly suppress the index \(K\), and simply write \(\mathcal{O}\) instead of \(\mathcal{O}^{\phantom{a}}_{K}\). Since we shall need some simple estimates later on to verify the condition (B3), we recall the following property for the absolute norm of non-zero elements in \(\mathcal{O}\), as defined by the index of \(x\mathcal{O}\) in \(\mathcal{O}\), so \(\mathrm{N}(x) = \mathrm{N}(x \mathcal{O}) \mathrel{\mathop{:}}=[\mathcal{O}: x \mathcal{O}]\).

The ring \(\mathcal{O}\) is a lattice in \(\mathbb{C}\simeq \mathbb{R}^2\) when \(d<0\), while the lattice property requires a Minkowski embedding for \(d>1\). For our discussion of power-free integers, it is thus convenient to consider imaginary and real quadratic fields separately, as we shall now do.

5.1 Imaginary quadratic fields↩︎

If \(K\) is an imaginary quadratic field, \(\mathcal{O}= \mathcal{O}^{\phantom{a}}_K\) is a lattice in \(\mathbb{C}\simeq \mathbb{R}^2\), such as the Gaussian integers \(\mathbb{Z}[\mathrm{i}]\) from \(K=\mathbb{Q}(\mathrm{i})\). Here, we identify \(z=x+\mathrm{i}y \in\mathbb{C}\) with the column vector \((x,y)^{\scriptscriptstyle\mathsf{T}} \! \in \mathbb{R}^2\) as usual. For our calculations, we now need a symmetric \(\mathbb{Q}\)-bilinear form on \(K\) that equals the standard Euclidean scalar product in \(\mathbb{R}^2\), which is \[\label{eq:form-complex} x.y \, \mathrel{\mathop{:}}=\, \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{2}} ({\overline{x }}y + x {\overline{y }}) \, = \, \mathrm{Re} ({\overline{x }} y) .\tag{16}\] The dual of \(\mathcal{O}\) then is \[\mathcal{O}^{*} \, = \, \{ y \in K : x.y \in \mathbb{Z}\text{ for all } x \in \mathcal{O}\} ,\] which is also the dual of \(\mathcal{O}\) in the sense of lattices in \(\mathbb{R}^2\), explaining the factor \(\frac{1}{2}\) in 16 .

Let us now write \(K = \mathbb{Q}(\mathrm{i}\sqrt{\delta}\,)\) with \(\delta \in \mathbb{N}\) square-free, which turns the original condition \(d\equiv 1 \bmod 4\) into \(\delta \equiv 3 \bmod 4\). Then, the following result is a special case of a more general property; see [35].

From here, one can calculate the dual of non-trivial principal ideals via \[\label{eq:dual-principal-ideal} (z)^{*} = \, (z \mathcal{O})^{*} = \, \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{{\overline{z }} }} \mathcal{O}^{*} .\tag{17}\] This allows to calculate the spectrum for all imaginary quadratic fields with class number \(1\), which are the nine [34] fields \(\mathbb{Q}(\mathrm{i}\sqrt{\delta}\,)\) with \[\delta \in \{ 1, 2, 3, 7, 11, 19, 43, 67, 163 \} .\] For the general case, we need the dual of an arbitrary non-trivial ideal, \(\mathfrak{a}\), which is given by \[\label{eq:dual-ideal-complex} \mathfrak{a}^{*} = \, {\overline{\mathfrak{a} }}^{-1} \mathcal{O}^{*}\tag{18}\] where \({\overline{\mathfrak{a} }}\) denotes the complex conjugate ideal and \(\mathfrak{b}^{-1} \mathrel{\mathop{:}}=\{ x\in K : x \mathfrak{b}\subseteq \mathcal{O}\}\) is the inverse of \(\mathfrak{b}\), and a fractional ideal. In particular, \(\mathfrak{b}^{-1} \mathfrak{b}= \mathcal{O}\). Since \(\mathcal{O}\) is a Dedekind domain, all non-trivial ideals are invertible; see [36] for background material. When \(\mathfrak{a}\) and \(\mathfrak{b}\) are coprime ideals, so \(\mathfrak{a}+ \mathfrak{b}= \mathcal{O}\), one has \[( \mathfrak{a}\mathfrak{b})^{*} = \, ( \mathfrak{a}\cap \mathfrak{b})^{*} = \, \mathfrak{a}^{*} + \mathfrak{b}^{*} = \, \bigl( {\overline{\mathfrak{a} }}^{-1} + {\overline{\mathfrak{b} }}^{-1} \bigr) \mathcal{O}^{*} = \, \mathfrak{c}^{-1} \mathcal{O}^{*}\] for a unique fractional ideal \(\mathfrak{c}\). The corresponding formula holds for the dual of multiple intersections, provided all ideals involved are pairwise coprime, which is the ideal-theoretic version of the CRT [35]. In this sense, we can always define the denominator via the fractional ideal \(\mathfrak{c}\), compare [35], and then decompose the denominator into prime ideals in the Dedekind domain \(\mathcal{O}\).

When viewing \(\mathcal{O}\) as a lattice in \(\mathbb{R}^2\), it is often useful to pin down a basis matrix for it in terms of the standard Cartesian basis of \(\mathbb{R}^2\). Here, for \(\delta \equiv 3 \bmod 4\), respectively for all other cases, natural choices are \[\label{eq:basis-imag} B \, = \, \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{2}} \begin{pmatrix} 2 & 1 \\ 0 & \sqrt{\delta} \end{pmatrix} \quad \text{and} \quad B \, = \, \begin{pmatrix} 1 & 0 \\ 0 & \sqrt{\delta} \end{pmatrix} .\tag{19}\] Let us discuss two quadratic fields of widespread interest, namely \(\delta = 1\) and \(\delta = 3\), before we continue with the general case.

5.1.1 Gaussian integers↩︎

Since \(\mathbb{Z}[\mathrm{i}] = \mathbb{Z}^2\), both arithmetically and geometrically, we begin with this case. Also, we profit from \(\mathbb{Z}[\mathrm{i}]\) being a Euclidean domain (and thus having unique prime factorisation up to units, that is, up to \(\{ 1, \mathrm{i}, -1, -\mathrm{i}\}\)).

Relative to \(\mathcal{O}\), there are three kinds of rational primes, namely \(2 = - \mathrm{i}(1+\mathrm{i})^2\), which is the only ramified prime, the primes \(p \equiv 3 \bmod 4\), which are inert and thus stay prime in \(\mathcal{O}\), and the splitting primes, \(p \equiv 1 \bmod 4\), giving complex conjugate pairs of Gaussian primes via \(p = \mathfrak{p}{\overline{\mathfrak{p} }}\), such as \(5=(2+\mathrm{i}) (2-\mathrm{i})\) or \(13=(3+2\mathrm{i})(3-2\mathrm{i})\). We thus get the Gaussian primes \[\mathcal{P}_{\!_{\mathrm{G}}}\, = \, \{ 1{+}\mathrm{i}, 3, \mathfrak{p}_{(5)}, {\overline{\mathfrak{p} }}_{(5)}, 7, 11, \mathfrak{p}_{(13)}, {\overline{\mathfrak{p} }}_{(13)}, \mathfrak{p}_{(17)}, {\overline{\mathfrak{p} }}_{(17)}, 19, \ldots \} .\] Here, they are listed along increasing rational primes and some rule for the splitting primes, such as taking a representative with positive real and imaginary parts and bigger real part first. Note that they are only unique up to units.

As a lattice in \(\mathbb{R}^2\), the ring \(\mathbb{Z}[\mathrm{i}]\) is self-dual, as follows from Lemma 1 because \(\delta = 1\) and \(\mathrm{i}\) is a unit. For any principal ideal \((z) = z \mathbb{Z}[\mathrm{i}]\) with \(z\ne 0\), we get its dual from 17 as \[(z)^{*} \, = \, \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{\bar{z}}} \mathbb{Z}[\mathrm{i}] \, = \, \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{|z|^2}} (z) ,\] which makes the calculations in this case particularly simple. Note that all principal ideals are square lattices again.

Now, let \(\boldsymbol{\kappa} = (\kappa_{\mathfrak{p}})^{\phantom{a}}_{\mathfrak{p}\in \mathcal{P}_{\!_{\mathrm{G}}}}\) with \(\kappa_{\mathfrak{p}} \in \mathbb{N}\cup\{\infty\}\), set \(\mathcal{P}_{\!_{\mathrm{G}}}' = \{ \mathfrak{p}\in \mathcal{P}_{\!_{\mathrm{G}}}: \kappa_{\mathfrak{p}} < \infty \}\) and define \[V_{{\scriptscriptstyle \mathrm{G}},\boldsymbol{\kappa}} \, = \, \mathbb{Z}[\mathrm{i}] \, \setminus \!\bigcup_{\mathfrak{p}\in\mathcal{P}_{\!_{\mathrm{G}}}'}\! \bigl( \mathfrak{p}^{\kappa_{\mathfrak{p}}}\bigr) ,\] with the special case \(V_{{\scriptscriptstyle \mathrm{G}}, \kappa}\) for constant \(\kappa_{\mathfrak{p}} = \kappa\), for all \(\mathfrak{p}\in\mathcal{P}_{\!_{\mathrm{G}}}\). For our further analysis, we need some restriction on \(\boldsymbol{\kappa}\), as indicated earlier. If \(\mathfrak{a}\subseteq \mathcal{O}\) is an ideal, we define \(\lambda (\mathfrak{a})\) from the condition (B3) for \(\mathfrak{a}\) viewed as a lattice, with respect to the Euclidean norm. We then need \[\label{eq:tail-cond} \sum_{\mathfrak{p}\in\mathcal{P}_{\!_{\mathrm{G}}}'} \lambda (\mathfrak{p}^{\kappa_{\mathfrak{p}}})^{-2} \, < \, \infty\tag{20}\] as our condition (B3). This can now be related to the norms of the ideals via Fact 14, then giving the slightly simpler sufficient condition \[\label{eq:G-cond} \sum_{\mathfrak{p}\in\mathcal{P}_{\!_{\mathrm{G}}}'} \mathrm{N}(\mathfrak{p})^{-\kappa^{\phantom{a}}_{\mathfrak{p}}} \, < \, \infty .\tag{21}\] To see this, observe that the field norm of \(z\in K\) is \(N (z) = z {\overline{z }}\) and agrees with the squared length of \(z\) when viewed as a vector in \(\mathbb{R}^2\). As the relation to the absolute norm is given by \(\mathrm{N}(z) = [ \mathcal{O}: z \mathcal{O}] = z {\overline{z }} = N (z)\), we can use Fact 14 for \(\lambda (\mathfrak{a})\).

Let us expand on the restrictions for the choice of \(\boldsymbol{\kappa}\). When \(\mathcal{P}_{\!_{\mathrm{G}}}'\) is a finite set, without any further restriction on the exponents \(\kappa^{\phantom{a}}_{\mathfrak{p}}\), the point set \(V^{\phantom{a}}_{{\scriptscriptstyle \mathrm{G}}, \boldsymbol{\kappa}}\) is lattice periodic, and the FB spectrum simply is the dual of the lattice of periods. Let us thus consider the case that \(\mathcal{P}_{\!_{\mathrm{G}}}'\) is an infinite set. Since only finitely many rational primes are ramified over \(K\), there are no restrictions on the corresponding exponents. When \(p\) is inert, the shortest non-zero element in \(p\mathcal{O}\) has length \(\lambda (p \mathcal{O}) = \sqrt{ \mathrm{N}(p \mathcal{O})} = p\), so that we only need \(\kappa^{\phantom{a}}_{p} \geqslant 1\) here. Finally, when \(p = \mathfrak{p}{\overline{\mathfrak{p} }}\) splits, we get \(\lambda (\mathfrak{p}) = \lambda ({\overline{\mathfrak{p} }}) = p\), which implies that the corresponding exponents need to be \(\geqslant 2\) for almost all of them, or for sufficiently many so that 21 and thus 20 holds.

The principal ideal \((\mathfrak{p}^{m} )\) is a square sublattice of \(\mathbb{Z}[\mathrm{i}]\) of index \[\bigl[ \mathbb{Z}[\mathrm{i}] : (\mathfrak{p}^m) \bigr] \, = \, \bigl[ \mathbb{Z}[\mathrm{i}] : \mathfrak{p}^m \mathbb{Z}[\mathrm{i}] \bigr] \, = \, \mathrm{N}(\mathfrak{p})^m ,\] from which one derives the natural density of \(V_{{\scriptscriptstyle \mathrm{G}},\boldsymbol{\kappa}}\) as \[\mathrm{dens}( V_{{\scriptscriptstyle \mathrm{G}}, \boldsymbol{\kappa}} ) \, = \prod_{\mathfrak{p}\in\mathcal{P}_{\!_{\mathrm{G}}}'} \bigl( 1 - \mathrm{N}(\mathfrak{p})^{-\kappa_{\mathfrak{p}}} \bigr) .\] This formula follows from an inclusion-exclusion argument via the periodic covering sets that are obtained by taking only finitely many primes into account, together with a tail estimate that relies on 21 , as in Section 4. The density formula simplifies to \(\mathrm{dens}( V_{{\scriptscriptstyle \mathrm{G}}, \kappa} ) = 1/\zeta^{\phantom{a}}_{\mathbb{Q}(\mathrm{i})} (\kappa)\) when all \(\kappa_{\mathfrak{p}}=\kappa\geqslant 2\), where \(\zeta^{\phantom{a}}_{\mathbb{Q}(\mathrm{i})}\) is the Dedekind zeta function of \(\mathbb{Q}(\mathrm{i})\).

To describe \(V_{{\scriptscriptstyle \mathrm{G}}, \boldsymbol{\kappa}}\) as a weak model set, we can now employ the CPS \((\mathbb{C}, H, \mathcal{L})\) with \(\mathbb{C}\simeq \mathbb{R}^2\) as direct space, the compact Abelian group \(H = \bigotimes_{\mathfrak{p}\in\mathcal{P}_{\!_{\mathrm{G}}}'} \mathbb{Z}[\mathrm{i}]/\mathfrak{p}^{\kappa_{\mathfrak{p}}} \mathbb{Z}[\mathrm{i}]\) as internal space, and the diagonal embedding of \(\mathbb{Z}[\mathrm{i}]\) into \(\mathbb{C}{\times} H\) as lattice, so \(\mathcal{L}= \bigl\{ (x,x^{\star}): x\in \mathbb{Z}[\mathrm{i}] \bigr\}\) with the \(\star\)-map being given by \(x\mapsto x^{\star} \mathrel{\mathop{:}}=(x \bmod \mathfrak{p}^{\kappa_{\mathfrak{p}}} )^{\phantom{a}}_{\mathfrak{p}\in \mathcal{P}_{\!_{\mathrm{G}}}'}\).

The FB spectrum of \(V^{\phantom{a}}_{{\scriptscriptstyle \mathrm{G}},\boldsymbol{\kappa}}\), and thus the dynamical spectrum of the induced MTDS with the measure \(\mu_{_\mathrm{M}}\), can now be calculated as in the previous section. Following the route from Remark 3, and observing \(\mathcal{O}^* = \mathcal{O}\) together with \(\bigl(\mathfrak{p}^{\kappa_{\mathfrak{p}}}\mathcal{O}\cap \mathfrak{q}^{\kappa_{\mathfrak{q}}} \mathcal{O}\bigr)^* = ({\overline{\mathfrak{p} }})^{-\kappa_{\mathfrak{p}}} + ({\overline{\mathfrak{q} }})^{-\kappa_{\mathfrak{q}}}\) for any pair \(\mathfrak{p}\ne \mathfrak{q}\) of Gaussian primes (and analogously for multiple intersections), one finds \[L^{\circledast} \, = \, \Bigl\{ k \in \mathbb{Q}[\mathrm{i}] : \begin{array}{c} \mathrm{den}(k) \text{ is (\kappa^{\phantom{a}}_{{\overline{\mathfrak{p} }}}{+} 1)-free for all } \mathfrak{p}\in \mathcal{P}_{\!_{\mathrm{G}}}' \\ \text{ and not divisible by any } \mathfrak{q}\in \mathcal{P}_{\!_{\mathrm{G}}}\!\!\setminus\! \mathcal{P}_{\!_{\mathrm{G}}}' \end{array} \Bigr\} \, = \, \sum_{\mathfrak{p}\in\mathcal{P}_{\!_{\mathrm{G}}}'} \bigl( \mathfrak{p}^{-\kappa^{\phantom{a}}_{{\overline{\mathfrak{p} }} }}\bigr) ,\] where \(L^{\circledast}\!/\mathcal{O}= \bigoplus_{\mathfrak{p}\in\mathcal{P}_{\!_{\mathrm{G}}}} \bigl( \mathfrak{p}^{-\kappa^{\phantom{a}}_{{\overline{\mathfrak{p} }} }}\bigr) /\mathcal{O}\) is the direct sum of countably many finite Abelian groups, all viewed as subgroups of \(\mathbb{Q}(\mathrm{i})/\mathcal{O}\subset \mathbb{T}^2\). This structure is also reflected in the CPS and its dual, the latter being \((\mathbb{C}, \widehat{H}, \mathcal{L}^{0})\) with \(\widehat{\mathbb{C}}\simeq \mathbb{C}\), the compact Abelian group \(\widehat{H} = \bigoplus_{\mathfrak{p}\in\mathcal{P}_{\!_{\mathrm{G}}}'} \mathcal{O}/ (\mathfrak{p}^{\kappa_{\mathfrak{p}}})\), where \(\mathcal{O}/ (\mathfrak{p}^{\kappa_{\mathfrak{p}}} ) \simeq (\mathfrak{p}^{-\kappa_{\mathfrak{p}}})/\mathcal{O}\), and the annihilator \(\mathcal{L}^{0}\) of \(\mathcal{L}\) as its lattice. Each of the contributing Abelian groups can be decomposed via Fact 13.

5.1.2 Eisenstein integers↩︎

With \(\xi = \frac{1}{2}(1 + \mathrm{i}\sqrt{3}\,)\), which is a primitive sixth root of unity, the ring of Eisenstein integers is \(\mathbb{Z}[\xi]\), which is the maximal order of \(K=\mathbb{Q}(\mathrm{i}\sqrt{3}\,)\), another Euclidean domain. The main difference to the previous example is the fact that \(\mathbb{Z}[\xi]\), viewed geometrically, is a triangular lattice in \(\mathbb{R}^2\). All principal ideals are then triangular lattices as well. The first step will be parallel to the above, but we will then take another step to change it into a square lattice for direct comparison with the \(\mathbb{Z}^2\)-action of symbolic dynamics.

The only ramified prime is \(p=3\), where one has \(3 = {\overline{\xi }} (1+\xi)^2\). The inert primes are the ones with \(p\equiv 2 \bmod 3\), while \(p\equiv 1 \bmod 3\) are the splitting primes, then with \(p = \mathfrak{p}{\overline{\mathfrak{p} }}\), such as \(7 = (2+\xi)(2+{\overline{\xi }})\) or \(13=(3+\xi)(3+{\overline{\xi }})\). The Eisenstein primes are thus \[\mathcal{P}_{_\mathrm{E}} \, = \, \{ 2, 1+\xi, 5, \mathfrak{p}^{\phantom{a}}_{(7)}, {\overline{\mathfrak{p} }}^{\phantom{a}}_{(7)}, 11, \mathfrak{p}^{\phantom{a}}_{(13)}, {\overline{\mathfrak{p} }}^{\phantom{a}}_{(13)}, 17, \mathfrak{p}^{\phantom{a}}_{(19)}, {\overline{\mathfrak{p} }}^{\phantom{a}}_{(19)}, 23, \ldots \} ,\] where we have \[\label{eq:Eisen-dual} \mathcal{O}^{*} = \, \frac{\raisebox{-2pt}{2 \mathrm{i}}}{\raisebox{0.5pt}{\sqrt{3}}} \mathcal{O}\quad \text{and} \quad (z)^{*} = \, \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{{\overline{z }}}} \mathcal{O}^{*} = \, \frac{\raisebox{-2pt}{2 \mathrm{i}z}}{\raisebox{0.5pt}{\sqrt{3} |z|^2}} \mathcal{O}.\tag{22}\] The required restrictions on \(\boldsymbol{\kappa}\) are completely analogous to the case of the Gaussian integers. With \({\mathcal{P}_{\!_{\mathrm{E}}}}{\!'} = \{ \mathfrak{p}\in \mathcal{P}_{\!_{\mathrm{E}}}: \kappa_{\mathfrak{p}} < \infty \}\), this then results in the FB spectrum \[L^{\circledast} = \, \frac{\raisebox{-2pt}{2\mathrm{i}}}{\raisebox{0.5pt}{\sqrt{3}}}\Bigl\{ k \in \mathbb{Q}(\xi) : \begin{array}{c} \mathrm{den}(k) \text{ is (\kappa^{\phantom{a}}_{{\overline{\mathfrak{p} }}} +1)-free for all \mathfrak{p}\in \mathcal{P}_{\!_{\mathrm{E}}}'} \\ \text{and not divisible by any \mathfrak{q}\in \mathcal{P}_{\!_{\mathrm{E}}}\!\setminus \mathcal{P}_{\!_{\mathrm{E}}}'} \end{array} \Bigr\} \, = \, \frac{\raisebox{-2pt}{2\mathrm{i}}}{\raisebox{0.5pt}{\sqrt{3}}} \sum_{\mathfrak{p}\in\mathcal{P}_{\!_{\mathrm{E}}}'} \bigl( \mathfrak{p}^{-\kappa^{\phantom{a}}_{{\overline{\mathfrak{p} }}}} \bigr) ,\] again computed via the dual lattice approach of Remark 3, and the formula from 22 . Like before, we also get the structure from Remark 4, here with \[L^{\circledast} \! / \mathcal{O}^{*} = \, \frac{ 2 \mathrm{i}}{\sqrt{3} } \bigoplus_{\mathfrak{p}\in\mathcal{P}_{\!_{\mathrm{E}}}'} \bigl( \mathfrak{p}^{-\kappa^{\phantom{a}}_{{\overline{\mathfrak{p} }}}} \bigr)/\mathcal{O},\] which emerges from the restriction of \(L^{\circledast}\) to the fundamental domain for \(\mathcal{O}^{*}\). Again, the representation chosen matches the needs of the dynamical context, while all contributing groups, up to isomorphism, can be decomposed via Fact 13.

Next, we want to adjust this result for a change from \(\mathcal{O}\) to a square lattice. Observe that a lattice \(\varGamma\subset \mathbb{R}^2\) with basis matrix \(B\) is turned into the standard integer lattice by left multiplication with \(B^{-1}\), so \(B^{-1} \varGamma= \mathbb{Z}^2\). Then, the dual is given by \[(B^{-1} \varGamma)^{*} = \, B^{\scriptscriptstyle\mathsf{T}} \varGamma^{*} .\] Here, the basis matrix of \(\mathcal{O}\) relative to the standard Cartesian basis reads \[B \, = \, \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{2}} \begin{pmatrix} 2 & 1 \\ 0 & \sqrt{3} \end{pmatrix} .\] Consequently, if we replace \(\varGamma= \mathcal{O}\) by \(B^{-1} \varGamma\), the FB spectrum changes to \(B^{\scriptscriptstyle\mathsf{T}} L^{\circledast}\), again with \(z=x+\mathrm{i}y\) being identified with the column vector \((x,y)^{\scriptscriptstyle\mathsf{T}}\).

5.1.3 General imaginary quadratic field↩︎

Let \(K\) be a general imaginary quadratic field, with maximal order \(\mathcal{O}= \mathcal{O}^{\phantom{a}}_{K}\). We can always view \(\mathcal{O}\) as a lattice in \(\mathbb{R}^2 \simeq \mathbb{C}\). Here, as the class number need not be \(1\), we have to work with prime ideals and their powers. We simply write \(\mathfrak{p}\) for a prime ideal, and \(\mathcal{P}_{\mathcal{O}}\) for the set of all prime ideals. Given \(\kappa_{\mathfrak{p}} \in\mathbb{N}\cup \{ \infty \}\) for \(\mathfrak{p}\in\mathcal{P}_{\mathcal{O}}\) subject to the condition \[\label{eq:gen-cond} \sum_{\mathfrak{p}\in\mathcal{P}^{\phantom{a}}_{\!\mathcal{O}}} [\mathcal{O}: \mathfrak{p}]^{-\kappa_{\mathfrak{p}}} \, < \, \infty ,\tag{23}\] we define \(V = \mathcal{O}\setminus \bigcup_{\mathfrak{p}\in\mathcal{P}^{\prime}_{\!\mathcal{O}}} \mathfrak{p}^{\kappa_{\mathfrak{p}}}\), where \(\mathcal{P}^{\prime}_{\!\mathcal{O}} = \{ \mathfrak{p}\in \mathcal{P}^{\phantom{a}}_{\! \mathcal{O}} : \kappa_{\mathfrak{p}} < \infty \}\). The denominator of a number \(k\in K\) is defined as before, now with coprimality understood in terms of ideals.

Turning \(\mathcal{O}\) into \(\mathbb{Z}^2\) as explained before, we get the corresponding result for the symbolic dynamical system as follows.

5.2 Real quadratic fields↩︎

Unlike for imaginary fields, the maximal order \(\mathcal{O}= \mathcal{O}^{\phantom{a}}_{K}\) of a real quadratic field \(K = \mathbb{Q}(\sqrt{d}\,)\) is a dense subset of \(\mathbb{R}\), but not a lattice. It becomes one under the standard Minkowski embedding \(\theta \colon \mathcal{O}\xrightarrow{\quad} \mathbb{R}^2\), as defined by \(x \mapsto (x, x')\) with \({}'\) being algebraic conjugation in \(K\). The latter is the unique field automorphism that fixes \(\mathbb{Q}\) and sends \(\sqrt{d}\mapsto - \sqrt{d}\). This gives \(\mathcal{L}= \theta (\mathcal{O}) = \{ (x, x') : x \in \mathcal{O}\}\), which is a lattice in \(\mathbb{R}^2\). Clearly, the mapping \(\theta\) is also well defined on \(K\).

To continue, we need the symmetric \(\mathbb{Q}\)-bilinear form \[\label{eq:bilinear-real} x.y \, \mathrel{\mathop{:}}=\, \mathrm{tr}(xy) \, = \, xy + x' y'\tag{24}\] which equals the standard scalar product of \(\theta (x)\) with \(\theta(y)\) in $^2 \(. It is related to the field norm\)N(x) = x x’$ via the estimate \[x.x \, \geqslant \, 2 \lvert N (x) \rvert ,\] as follows from \(x.x = (x \mp x' )^2 \pm 2 x x'\) via a case distinction whether \(xx'\) is positive or negative. This will later again allow us to use Fact 14.

Relative to 24 , the dual of \(\mathcal{O}\) reads \[\mathcal{O}^* \, = \, \{ y \in K : x.y \in \mathbb{Z}\text{ for all } x \in \mathcal{O}\} ,\] which agrees with the co-different of \(\mathcal{O}\) relative to \(\mathbb{Z}\) here and can be calculated as a special case of [35]. In our setting, this can be stated as follows.

Let us mention in passing that the basis matrices of \(\theta (\mathcal{O})\) with respect to the standard Cartesian basis of \(\mathbb{R}^2\) read \[\label{eq:basis-real} B \, = \, \begin{pmatrix} 1 & \sqrt{d}\, \\ 1 & {-}\sqrt{d} \end{pmatrix} \quad \text{and} \quad B \, = \, \begin{pmatrix} 1 & \frac{1+\sqrt{d}}{2}\, \\ 1 & \frac{1-\sqrt{d}}{2} \end{pmatrix}\tag{25}\] for \(d\not \equiv 1 \bmod 4\) and \(d\equiv 1 \bmod 4\), respectively.

Now, Lemma 2 also gives us the dual of any non-trivial principal ideal, namely \[\label{eq:gen-id} (x)^* = \, \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{x}}\mathcal{O}^* = \, \frac{\raisebox{-2pt}{\psi^{\phantom{a}}_{d}}}{\raisebox{0.5pt}{x}} \mathcal{O}\, = \, \frac{\raisebox{-2pt}{\psi^{\phantom{a}}_{d} }}{\raisebox{0.5pt}{N(x)}} (x') ,\tag{26}\] where \(\psi^{\phantom{a}}_{d} = 1/\sqrt{d}\) for \(d\equiv 1 \bmod 4\) and half of this number for all other cases. For a general non-trivial ideal \(\mathfrak{a}\), we get the dual as \[\mathfrak{a}^{*} = \, \mathfrak{a}^{-1} \mathcal{O}^{*} \, = \, \psi^{\phantom{a}}_{d} \, \mathfrak{a}^{-1} \mathcal{O},\] again with the inverse ideal \(\mathfrak{a}^{-1}\).

Cases with class number \(1\) (in the wider sense) include \(\mathbb{Q}(\sqrt{d}\,)\) with \[d \in \{ 2^*, 3, 5^*, 6, 7, 11, 13^*, 14, 17^*, 19, 21, 22, 23, 29^*, 31, ... \} ,\] where it is still unknown whether this set is infinite (as conjectured) or not. Integers with a \({}^*\) mark cases whose class number is \(1\) also in the narrow sense [34]; see Sequences A003172 and A003655 of [40]. Let us consider two special cases first.

5.2.1 The case \(d=2\)↩︎

Let us treat \(K=\mathbb{Q}(\sqrt{2}\,)\) and its maximal order, \(\mathcal{O}=\mathbb{Z}[\sqrt{2}\,]\), which is one of the cases with class number \(1\). Lemma 2 and Eq. 26 give \[\mathcal{O}^{*} = \, \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{2 \sqrt{2}}} \mathcal{O}\, = \, \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{4}} (\sqrt{2} \, ) \quad \text{and} \quad (x)^{*} = \, \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{2 \sqrt{2}x}} \mathcal{O}.\]

The primes of \(\mathcal{O}\) are again of three types and emerge from the rational primes as follows. There is one ramified prime, which is \(2 = (\sqrt{2}\,)^2\), while all rational primes \(p\equiv \pm 3 \bmod 8\) are inert. The splitting primes are the rational primes with \(p \equiv \pm 1 \bmod 8\), and any such prime splits as \(p = (r + s \sqrt{2} \, )(r - s \sqrt{2}\,)\), so \(p=r^2 - 2 s^2\), with coprime \(r,s \in \mathbb{Z}\). We thus get the list of primes in \(\mathcal{O}\) as \[\mathcal{P}^{\phantom{a}}_{\mathcal{O}} \, = \, \{ \sqrt{2}, 3, 5, \mathfrak{p}^{\phantom{a}}_{(7)},\mathfrak{p}^{\prime}_{(7)}, 11, 13, \mathfrak{p}^{\phantom{a}}_{(17)}, \mathfrak{p}^{\prime}_{(17)}, 19, \ldots \} ,\] again listed along increasing rational primes. Note that the absolute norm for elements of \(K\) is given by \(\mathrm{N}( a + b \sqrt{2}\, ) = \lvert a^2-2 b^2 \rvert\).

With \(\mathcal{P}\) and \(\mathcal{P}'\) defined in analogy to above, the FB spectrum is \[L^{\circledast} = \, \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{2\sqrt{2}}} \Bigl\{ k \in K : \! \begin{array}{c} \mathrm{den}(k) \text{ is (\kappa_{\mathfrak{p}} + 1)-free for all \mathfrak{p}\in \mathcal{P}^{\prime}_{\mathcal{O}}} \\ \text{and not divisible by any \mathfrak{q}\in \mathcal{P}^{\phantom{a}}_{\mathcal{O}} \! \setminus\mathcal{P}^{\prime}_{\mathcal{O}}} \end{array} \! \Bigr\} \, = \, \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{2\sqrt{2}}} \sum_{\mathfrak{p}\in \mathcal{P}^{\prime}_{\mathcal{O}}} \bigl( \mathfrak{p}^{-\kappa_\mathfrak{p}} \bigr) ,\] again calculated via the dual lattices and Remark 3. One can now turn \(\theta (\mathcal{O})\) into \(\mathbb{Z}^2\) by left multiplication with \(B^{-1}\) as before, and the FB spectrum then becomes \(B^{\scriptscriptstyle\mathsf{T}} L^{\circledast}\).

5.2.2 The case \(d=5\)↩︎

For \(K = \mathbb{Q}(\sqrt{5}\,)\), one has \(\mathcal{O}=\mathbb{Z}[\tau]\), where \(\tau = \frac{1}{2}(1{+}\sqrt{5}\,)\) is the golden ratio. Here, Lemma 2 and Eq. 26 give \[\mathcal{O}^{*} = \, \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{\sqrt{5}}} \mathcal{O}\, = \, \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{5}} (\sqrt{5}\,) \quad \text{and} \quad (x)^{*} = \, \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{\sqrt{5}x}} \mathcal{O}.\] The primes of \(\mathcal{O}\) emerge from \(\mathcal{P}\) as \(5 = (2 \tau {-} 1)^2\), which is ramified, while any \(p \equiv \pm 2 \bmod 5\) is inert and \(p\equiv \pm 1 \bmod 5\) splits, the latter as \(p = (r + s \tau) (r + s \tau' )\), so \(p = r^2 + rs - s^2\), hence \[\mathcal{P}^{\phantom{a}}_{\mathcal{O}} \, = \, \{ 2, 3, \sqrt{5}, 7, \mathfrak{p}^{\phantom{a}}_{(11)}, \mathfrak{p}^{\prime}_{(11)}, 13, 17, \mathfrak{p}^{\phantom{a}}_{(19)}, \mathfrak{p}^{\prime}_{(19)}, \ldots \} .\] The rest is completely analogous to the previous example.

5.2.3 General real quadratic field↩︎

The general situation is analogous to the imaginary case, except that we need the Minkowski embedding \(\theta\) to obtain lattices in \(\mathbb{R}^2\). We can thus state the result as follows.

The situation after changing to (geometric) \(\mathbb{Z}^2\)- and \(\mathbb{R}^2\)-action is as follows.

This corollary illustrates the role of the explicit representation for the dynamics. Up to group isomorphism, we get the same spectra as in Theorem 18, but the representations are rather different. This is precisely what the Halmos–von Neumann theorem uses to distinguish ergodic measure-theoretic dynamical systems with pure-point spectrum up to measurable conjugacy. When they fail to be conjugate, this can be caused by a linear transform (as here), but it can also be more substantial (and thus more relevant).

While we have treated a particular class of examples, the structure is sufficiently robust to allow for various generalisations. Also, the role of different ergodic measures can certainly be explored further, where we expect the FB coefficients to feature prominently. It is presently open to what extent the more detailed results can be transferred from the number-theoretic setting to the much more general lattice setting.

6 Bernoulli thinning of uniformly distributed sequences↩︎

The purpose of this appendix is a brief, self-contained exposition of the equidistribution result for random subsequences of a uniformly distributed sequence, adapted to our needs in the main text. The result as such is well known, but not so easy to locate in the literature; see [41] for general background and [42] (and references therein) for a more specific treatment in the context of locally compact groups.

Below, we shall first use the standard approach to uniform distribution of sequences in \([0,1]\), and then its generalisation to bounded sets \(U\subset \mathbb{R}\) (or \(U \subset \mathbb{R}^d\)) such that \(\mathbf{1}^{\phantom{a}}_{U}\) is Riemann integrable, which is the case if and only if \(\partial U\) has zero Lebesgue measure. Such sets are also called Jordan measurable. There is a well-known generalisation to (relatively) compact subsets of second countable, locally compact Abelian groups, but we shall also need uniform distribution in compact sets with a ‘fat’ boundary (of positive measure).

Let us begin with the unit interval, compare [41], and let the sequence \((a^{\phantom{a}}_{m})^{\phantom{a}}_{m\in \mathbb{N}}\) be uniformly distributed in \([0,1]\), which is to say that, for all \(0\leqslant \alpha < \beta \leqslant 1\), one has \[\label{app:eq-1} \lim_{n\to\infty} \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{n}} \sum_{m=1}^{n} \boldsymbol{1}^{\phantom{a}}_{[\alpha,\beta]} (a^{\phantom{a}}_{m}) \, = \, \beta-\alpha .\tag{27}\] Now, we are interested in the question what happens if we go to a subsequence \(\bigl( a^{\phantom{a}}_{m_i}\bigr)_{i\in\mathbb{N}}\) that emerges from a random selection of elements. The standard formulation employs a family \((\xi^{\phantom{a}}_{i})^{\phantom{a}}_{i\in\mathbb{N}}\) of i.i.d.Bernoulli random variables, which take values in \(\{ 0,1\}\) with common distribution defined by \(\mathbb{P}(\xi^{\phantom{a}}_{1} = 1) = p \in (0,1)\). Then, each realisation gives rise to such a subsequence by keeping the \(a^{\phantom{a}}_{m}\) where \(\xi^{\phantom{a}}_{m} =1\). We call this Bernoulli thinning. We now need the strong law of large numbers (SLLN) in several forms.

Let \(0\leqslant \alpha < \beta \leqslant 1\) be arbitrary, but fixed. One can define a new family of random variables via \(X^{\phantom{a}}_m = \xi^{\phantom{a}}_m \boldsymbol{1}^{\phantom{a}}_{[\alpha,\beta]} (a^{\phantom{a}}_{m})\), which are independent, but no longer identically distributed. Each \(X_n\) is either identically \(0\) or takes values in \(\{ 0, 1\}\), so the variance is always bounded by \(1\). Then, Kolmogorov’s version of the SLLN applies, compare [27], and we get \[\lim_{n\to\infty} \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{n}} \sum_{\ell=1}^{n} \bigl( X^{\phantom{a}}_{\ell} - \mathbb{E}(X^{\phantom{a}}_{\ell}) \bigr) \, = \, 0 ,\] which holds almost surely, that is, for almost all realisations of the thinning. Since \(\mathbb{E}(X^{\phantom{a}}_{\ell}) = p\) if \(a^{\phantom{a}}_{\ell} \in [\alpha, \beta]\) and \(\mathbb{E}(X^{\phantom{a}}_{\ell}) = 0\) otherwise, using 27 , one has \[\frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{n}} \sum_{\ell=1}^{n} \mathbb{E}(X^{\phantom{a}}_{\ell}) \, = \, \frac{\raisebox{-2pt}{p}}{\raisebox{0.5pt}{n}} \sum_{\ell=1}^{n} \boldsymbol{1}^{\phantom{a}}_{[\alpha, \beta]} (a^{\phantom{a}}_{\ell}) \, \xrightarrow{\, n\to\infty\,} \, p (\beta - \alpha) .\] Observing that \(\frac{1}{n} \sum_{i=1}^{n} \xi^{\phantom{a}}_{i}\) almost surely converges to \(p\), by the standard version of the SLLN, we get, almost surely as \(n\to\infty\), the asymptotic behaviour \[\frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{n}} \sum_{m=1}^{n} X^{\phantom{a}}_{m} \, \sim \, \frac{p}{\sum_{i=1}^{n} \xi^{\phantom{a}}_{i}} \sum_{\substack{m=1_{\vphantom{a}} \\ \xi^{\phantom{a}}_{m} = 1}}^{n} \boldsymbol{1}^{\phantom{a}}_{[\alpha,\beta]} (a^{\phantom{a}}_{m}) \, = \, \frac{\raisebox{-2pt}{p}}{\raisebox{0.5pt}{\#^{\phantom{a}}_n}} \sum_{i=1}^{\#^{\phantom{a}}_n} \boldsymbol{1}^{\phantom{a}}_{[\alpha,\beta]} (a^{\phantom{a}}_{m_i}) ,\] where \(\#^{\phantom{a}}_n\) is the (random) number of elements of the original sequence \((a^{\phantom{a}}_{1}, a^{\phantom{a}}_{2}, \ldots , a^{\phantom{a}}_{n})\) that remain after the Bernoulli thinning and \((m_i)^{\phantom{a}}_{i\in\mathbb{N}}\) is the sequence of integers for which \(\xi^{\phantom{a}}_{m_i}=1\). Note that, almost surely as \(n\to\infty\), one has \(\#^{\phantom{a}}_n \sim p n\), again by the SLLN.

Putting the two pieces together, we see that \[\lim_{N\to\infty} \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{N}} \sum_{i=1}^{N} \boldsymbol{1}^{\phantom{a}}_{[\alpha, \beta]} (a^{\phantom{a}}_{m_i}) \, = \, \beta - \alpha ,\] which holds almost surely in the above sense.

By a standard countability argument, we then also know the simultaneous almost sure validity of the previous limit for all \(\alpha, \beta \in \mathbb{Q}\cap [0,1]\), which is dense in \([0,1]\).

Finally, monotonicity in conjunction with inner and outer regularity of Lebesgue measure gives us the almost sure equidistribution as follows.

It is well known [41] that the limit in Eq. 27 , by the regularity of Lebesgue measure \(\lambda_{_\mathrm{L}}\), has the generalisation \[\lim_{n\to\infty} \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{n}} \sum_{\ell=1}^{n} \boldsymbol{1}^{\phantom{a}}_{W} (a^{\phantom{a}}_{\ell}) \, = \, \lambda_{_\mathrm{L}} (W) ,\] to any Jordan-measurable set \(W\subseteq [0,1]\), and, more generally, also to any Jordan-measurable set \(W\subseteq [0,1]^d\subset \mathbb{R}^d\), when the sequence \((a^{\phantom{a}}_n)^{\phantom{a}}_{n\in\mathbb{N}}\) is uniformly distributed in \([0,1]^d\); consult [41] for further details.

There is a generalisation of the entire concept to compact subsets of a locally compact Abelian group \(G\) that is second countable (which means that is has a countable basis for its topology); see [41] as well as [42] and references therein. To keep things simple, we note that we only need this generalisation for compact Abelian groups of the product form \(H = \bigotimes_{i\in\mathbb{N}} G_i\) where each \(G_i\) is finite Abelian. Here, second countability of \(H\) follows constructively. Indeed, since each \(G_i\) has only finitely many subsets and since the finite subsets of \(\mathbb{N}\) are countable, the collection \[\bigl\{ (U_i)_{i\in\mathbb{N}} : U_i \subseteq G_i \text{ with } U_i = G_i \text{ for all but finitely many } i \bigr\}\] defines a countable basis for the topology of \(H\). We abbreviate the product set of the \(U_i\) as \((U^{\phantom{a}}_{1}, U^{\phantom{a}}_{2}, \ldots , U^{\phantom{a}}_{N})\) if \(U_i = G_i\) holds for all \(i > N\), and agree to use the smallest index \(N\) with this property. These sets play the role of cylinder sets.

We now assume \(G\) to be equipped with its normalised Haar measure \(\nu_{_\mathrm{H}}\), so \(\nu^{\phantom{a}}_{H} (H) =1\). Clearly, we then get the measure of a cylinder set as \[\nu_{_\mathrm{H}}(U^{\phantom{a}}_{1}, U^{\phantom{a}}_{2}, \ldots , U^{\phantom{a}}_{N}) \, = \prod_{i=1}^{N} \frac{| U_i |}{| G_i |} ,\] where \(|.|\) denotes the cardinality of a finite set.

We can now define uniform distribution of a sequence \((a_n)^{\phantom{a}}_{n\in\mathbb{N}}\) in \(H\) or in compact subsets of \(H\) as follows.

Then, our previous thinning argument can be applied to any set of the form \(W\cap C\), of which there are countably many, and we get the following result.

It is clear that several immediate generalisations are possible for going beyond finite factors, but we leave further details in this direction to the interested reader.

If we are in the situation of Corollary 21, we know that the sequence of probability measures defined by \(\frac{1}{n} \sum_{j=1}^{n} \delta_{a_j}\) weakly converges to the probability measure \(\frac{1}{\nu_{_\mathrm{H}}(W)} \, \nu_{_\mathrm{H}}\big|_{W}\).

Uniform distribution of a sequence \((a_m)^{\phantom{a}}_{m\in\mathbb{N}}\) in a compact set \(W\) means that \(\frac{1}{n}\sum_{m=1}^{n} a_m\), as \(n\to\infty\), converges to \(\int_{W} \,\mathrm{d}\nu_{_\mathrm{H}}= \nu_{_\mathrm{H}}(W) = \nu_{_\mathrm{H}}(\boldsymbol{1}^{\phantom{a}}_{W})\). More generally, for every continuous function \(f\) on \(H\), we also get \[\lim_{n\to\infty} \frac{\raisebox{-2pt}{1}}{\raisebox{0.5pt}{n}} \sum_{m=1}^{n} f (a_m) \, = \int_{W} f(x) \,\mathrm{d}\nu_{_\mathrm{H}}(x) ,\] provided the sequence is uniformly distributed in \(W\). In particular, this holds for all characters on \(H\), which are the continuous group homomorphisms from \(H\) into the unit circle. This result then allows to determine the FB coefficients of model sets, including weak model sets of maximal density like the ones in this paper, on the basis of the result from [21].

Acknowledgements↩︎

It is our pleasure to thank Philipp Gohlke, Markus Kirschmer, Christoph Richard, Nicolae Strungaru and Vitali Wachtel for helpful discussions, as well as Fabian Gundlach, Jürgen Klüners, Neil Mañibo, Jan Mazáč, Andreas Nickel and Timo Spindeler for useful hints on the manuscript.

This work was supported by the German Research Council (Deutsche Forschungsgemeinschaft, DFG) under SFB-TRR 358/1 (2023) – 491392403 (MB, DL) and by the Austrian Science Fund FWF: P-33943-N (TS), by a grant from the priority research area SciMat under the Strategic Programme Excellence Initiative at Jagiellonian University (TS), and by the MSCA individual fellow project ErgodicHyperbolic - 101151185 (TS).

References↩︎

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1. More generally, the dynamical point spectrum is the smallest group that contains the FB spectrum.↩︎