Aperiodic linearly repetitive Delone sets are densely repetitive
英文摘要:
We show that aperiodic linearly repetitive Delone sets are densely repetitive. This confirms a conjecture of Lagarias and Pleasants.
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Daniel Lenz
We show that aperiodic linearly repetitive Delone sets are densely repetitive. This confirms a conjecture of Lagarias and Pleasants.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
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We extend a discrepancy bound of Lagarias and Pleasants for local weight distributions on linearly repetitive Delone sets and show that a similar bound holds also for the more general case of Delone sets without finite local complexity if linear repetitivity is replaced by $\varepsilon$-linear repetitivity. As a result we establish that Delone sets that are $\varepsilon$-linear repetitive for some sufficiently small $\varepsilon$ are rectifiable, and that incommensurable multiscale substitution tilings are never almost linearly repetitive.
Two Delone sets are bounded distance equivalent to each other if there is a bijection between them such that the distance of corresponding points is uniformly bounded. Bounded distance equivalence is an equivalence relation. We show that the hull of a repetitive Delone set with finite local complexity has either one equivalence class or uncountably many. A very similar result is proven in arXiv:2011.00106 [math.MG].
Non-periodic systems have become more important in recent years, both theoretically and practically. Their description via Delone sets requires the extension of many standard concepts of crystallography. Here, we summarise some useful notions of symmetry and aperiodicity, with special focus on the concept of the hull of a Delone set. Our aim is to contribute to a more systematic and consistent use of the different notions.
We construct examples of Delone sets of the plane (that is, discrete subsets that are uniformly separated and coarsely dense) that are repetitive (each patch of the set appears in every large-enough ball) though non-rectifiable (i.e. non bi-Lipschitz equivalent to the standard lattice). More generally, we construct such a set so that the translation action on the closure of its orbit has any prescribed Choquet simplex as its set of invariant probability measures (in particular, we provide uniquely ergodic examples). The construction relies on classical examples of Burago-Kleiner and McMullen, …
In this paper we prove that, for any integer $d>0$, every linearly repetitive Delone set in the Euclidean $d$-space $\RR^d$ is equivalent, up to a bi-Lipschitz homeomorphism, to the integer lattice $\ZZ^d$. In the particular case when the Delone set $X$ in $\RR^d$ comes from a primitive substitution tiling of $\RR^d$, we give a condition on the eigenvalues of the substitution matrix which implies the existence of a homeomorphism with bounded displacement from $X$ to the lattice lattice $λ\ZZ^d$ for some positive $λ$. This condition includes primitive Pisot substitution tilings but also concern…
Linearly repetitive Delone sets are shown to be rectifiable by a bi-Lipschitz homeomorphisms of the Euclidean space that sends the Delone set to the set of points with integer coordinates.
BiLipschitz and bounded displacement equivalence of Delone sets
本文综述Delone集的双Lipschitz与有界位移等价性,重点研究非周期序相关实例,涉及经典结果、相关判据及替换铺砌、截集等类别的扩展情况。
On rectifiability of Delone sets in intermediate regularity
本研究运用相关技术给出Delone集等价于标准格的充分条件,推广McMullen等的结论,证明特定ω-重复Delone集可等价于标准格,并探讨规定体积形式方程的解以拓展相关成果。
Fast repetitivity in non-rectifiable Delone sets
针对d≥2的欧几里得空间,构造了非可矩形但具有重复性的Delone集,并得到了接近最优的重复性函数,证明基于Burago-Kleiner的编码不可实现密度的方法。
Let $X$ be a (repetitive) infinite connected simple graph with a finite upper bound $Δ$ on the vertex degrees. The main theorem states that $X$ admits a (repetitive) limit aperiodic vertex coloring by $Δ$ colors. This refines a theorem for finite graphs proved by Collins and Trenk, and by Klavžar, Wong and Zhu, independently. It is also related to a theorem of Gao, Jackson and Seward stating that any countable group has a strongly aperiodic coloring by two colors, and to recent research on distinguishing number of graphs by Lehner, Pilśniak and Stawiski, and by Hüning et al. In our theorem, th…
This paper addresses the problem of describing aperiodic discrete structures that have a self-similar or self-affine structure. Substitution Delone set families are families of Delone sets (X_1, ..., X_n) in R^d that satisfy an inflation functional equation under the action of an expanding integer matrix in R^d. This paper studies such functional equation in which each X_i is a discrete multiset (a set whose elements are counted with a finite multiplicity). It gives necessary conditions on the coefficients of the functional equation for discrete solutions to exist. It treats the case where the…
Sturmian lattices and Aperiodic tile sets
本文提出基于二次斜率斯特鲁米安词构造非周期性瓷砖集的算法,并利用斯特鲁米安晶格和Delone集的有界位移等价性进行分类和构造。
We define a general framework that includes objects such as tilings, Delone sets, functions and measures. We define local derivability and mutual local derivability (MLD) between any two of these objects in order to describe their interrelation. This is a generalization of the local derivability and MLD (or S-MLD) for tilings and Delone sets which are used in the literature, under a mild assumption. We show that several canonical maps in aperiodic order send an object P to one that is MLD with P. Moreover we show that, for an object P and a class S of objects, a mild condition on them assures …
In the study of aperiodic order and mathematical models of quasicrystals, questions regarding equivalence relations on Delone sets naturally arise. This work is dedicated to the bounded displacement (BD) equivalence relation, and especially to results concerning instances of non-equivalence. We present a general condition for two Delone sets to be BD non-equivalent, and apply our result to Delone sets associated with tilings of Euclidean space. First we consider substitution tilings, and exhibit a substitution matrix associated with two distinct substitution rules. The first rule generates onl…
This paper characterizes when a Delone set X is an ideal crystal in terms of restrictions on the number of its local patches of a given size or on the hetereogeneity of their distribution. Let N(T) count the number of translation-inequivalent patches of radius T in X and let M(T) be the minimum radius such that every closed ball of radius M(T) contains the center of a patch of every one of these kinds. We show that for each of these functions there is a `gap in the spectrum' of possible growth rates between being bounded and having linear growth, and that having linear growth is equivalent to …
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