Ergodic properties of visible lattice points

Baake, Michael; Huck, Christian

doi:10.1134/s0081543815010137

Cited by 25 publications

(6 citation statements)

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“…This also indicates that S has positive configurational entropy [28,49], which might seem to contradict pure point diffractivity. But S has in fact zero Kolmogorov-Sinai entropy with respect to the natural pattern frequency measure [6,7,31].…”

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confidence: 99%

A short guide to pure point diffraction in cut-and-project sets

Richard¹,

Strungaru²

2017

J. Phys. A: Math. Theor.

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We dedicate this work to Tony Guttmann on the occasion of his 70 th birthday.Abstract. We briefly review the diffraction of quasicrystals and then give an elementary alternative proof of the diffraction formula for regular cut-andproject sets, which is based on Bochner's theorem from Fourier analysis. This clarifies a common view that the diffraction of a quasicrystal is determined by the diffraction of its underlying lattice. To illustrate our approach, we will also treat a number of well-known explicitly solvable examples. OutlineQuasicrystals are highly ordered rigid structures that are -unlike crystalsintrinsically non-periodic. Their discovery by diffraction experiments on certain rapidly cooled alloys [60,30] in 1982 was a surprise, since a combination of the above two properties was regarded unphysical at that time. In fact quasicrystals have been recently found in nature [15,16]. In 2011 the Nobel Prize in chemistry was awarded to Dan Shechtman for the discovery of quasicrystals.A natural mathematical idealisation of quasicrystals are (regular) cut-and-project sets, i.e., projections of suitable lattice subsets from some higher-dimensional superspace. The Bragg part in the diffraction of a cut-and-project set had been rigorously computed by Hof [28] in 1995. Pure point diffractivity of a cut-and-project set was proved by Schlottmann [58] however only in 2000 (in fact for a more general setting than Euclidean space). It has been argued from the beginning that the diffraction of a cut-and-project set should be deducible from that of the underlying lattice. However a corresponding proof appeared only in 2013 by Baake and Grimm [4] 1 for Euclidean superspace. In fact the diffraction formula for a cut-and-project set is, in a certain sense, equivalent to the diffraction formula for the lattice in superspace, as has been shown recently [55]. The diffraction formula also holds for non-regular cut-and-project schemes of extremal density [7,32], and it may be used to describe the Bragg part in the diffraction of certain random cut-and-project sets [11,38,53].Whereas there are some excellent expository reviews on aperiodic order, see e.g. [2,3], it is our intention to actually re-derive a substantial part of the above

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confidence: 99%

A short guide to pure point diffraction in cut-and-project sets

Richard¹,

Strungaru²

2017

J. Phys. A: Math. Theor.

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“…We mention the monograph [3] for a modern comprehensive exposition of the subject. Recently, there has been a renewed interest in such non-regular model sets due to their rich combinatorial and dynamical properties, see [4,9] and references therein.…”

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confidence: 99%

On Pattern Entropy of Weak Model Sets

Huck

Richard

2015

Discrete Comput Geom

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“…They prove that these B-free systems are proximal. (II) B-free systems in number fields Baake and Huck [4] also define B-free integers in number fields which generalizes the case studied by Cellarosi and Vinogradov [8] and (I). Given a finite extension K of Q, with the ring of integers O K , they set F B :" O K z Ť bPB b, where B is an infinite pairwise coprime collection of ideals in O K with ř bPB 1 |O K {b| ă 8.…”

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confidence: 88%

“…This contradicts that elements of tΛ i u iě1 are pairwise coprime. 4. Proximality of pX η , pS a q aPO m K q.…”

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Proximality of multidimensional $ \mathscr{B} $-free systems

Dymek¹

2021

DCDS

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We characterize proximality of multidimensional B-free systems in the case of number fields and lattices in Z m , m ě 2.1. Introduction. In this paper, we study proximality of generalizations of B-free systems [1,10]. Let B Ď N. Integers with no factors in B are called B-free numbers and are denoted by F B . Such sets were studied already in the 30's by Behrend, Chowla, Davenport, Erdős, Schur and others. Note that, if S " tp 2 : p is primeu then 1 F S " µ 2 , where µ : Z Ñ C is the Möbius function given by the following formula:The dynamical approach to study B-free systems is rather new. In 2010, Sarnak in his seminal paper [17] proposed to study the dynamical systems determined by µ and µ 2 . In either case, we consider the closure X η of the orbit of η " µ P t´1, 0, 1u Z or η " µ 2 P t0, 1u Z under the left shift S. The dynamics of pX µ , Sq is complicated and there are many open questions related to it. The system pX µ 2 , Sq (called squarefree system) which is a topological factor of pX µ , Sq via the map px n q nPZ Þ Ñ px 2 n q nPZ is simpler to study. Similarly, given B Ď N, taking the closure of the orbit of η " 1 F B P t0, 1u Z under the left shift, yields a B-free system. At first, B-free systems were studied in the Erdős case, i.e., for B infinite, pairwise coprime, withLet G be a countable abelian group. Recall that proper subgroups H 1 , H 2 Ď G are said to be coprime whenever H 1`H2 " G. Let pT g q gPG be an action of G by homeomorphisms on a compact metric space pX, Dq. The pair pX, pT g q gPG q is called a topological dynamical system. Two complementary, basic concepts of topological dynamics are distality and proximality. Recall that a pair px, yq of points from X is called distal if lim inf gÑ8 DpT g x, T g yq ą 0, otherwise px, yq is called proximal. If any pair of distinct points in X is distal (respectively, proximal) then the

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Ergodic properties of visible lattice points

Cited by 25 publications

References 38 publications

A short guide to pure point diffraction in cut-and-project sets

A short guide to pure point diffraction in cut-and-project sets

On Pattern Entropy of Weak Model Sets

Proximality of multidimensional $ \mathscr{B} $-free systems

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