Statistics of patterns in typical cut and project sets

当切削与投影集是替换性的吗？

When is a cut and project set substitutional?

本文探讨了切削与投影集在何种条件下可通过替换规则定义，揭示了其与无理切片和晶格结构的关系。

Gaps problems and frequencies of patches in cut and project sets

We establish a connection between gaps problems in Diophantine approximation and the frequency spectrum of patches in cut and project sets with special windows. Our theorems provide bounds for the number of distinct frequencies of patches of size r, which depend on the precise cut and project sets being used, and which are almost always less than a power of log r. Furthermore, for a substantial collection of cut and project sets we show that the number of frequencies of patches of size r remains bounded as r tends to infinity. The latter result applies to a collection of cut and project sets o…

切割-投影集的有界距离等价性与等分解性

Bounded distance equivalence of cut-and-project sets and equidecomposability

本文研究了切割-投影集的有界距离等价性与等分解性关系，揭示了等分解性在确定有界距离等价类中的关键作用。

Relative topological entropy for actions of non-discrete groups on compact spaces in the context of cut and project schemes

In the study of aperiodic order via dynamical methods, topological entropy is an important concept. In this paper, parts of the theory, like Bowen's formula for fibre wise entropy or the independence of the definition from the choice of a Van Hove sequence, are extended to actions of several non-discrete groups. To establish these results, we will show that the Ornstein-Weiss lemma is valid for all considered groups which appear in the study of cut and project schemes.

On generalized self-similarities of cut-and-project sets

Cut-and-project sets $Σ\subset\mathbb{R}^n$ represent one of the types of uniformly discrete relatively dense sets. They arise by projection of a section of a higher-dimensional lattice to a suitably oriented subspace. Cut-and-project sets find application in solid state physics as mathematical models of atomic positions in quasicrystals, the description of their symmetries is therefore of high importance. We focus on the question when a linear map $A$ on $\mathbb{R}^n$ is a self-similarity of a cut-and-project set $Σ$, i.e.\ satisfies $AΣ\subsetΣ$. We characterize such mappings $A$ and provid…

Classification and statistics of cut and project sets

We define Ratner-Marklof-Strombergsson measures. These are probability measures supported on cut-and-project sets in R^d (d > 1) which are invariant and ergodic for the action of the groups ASL_d(R) or SL_d(R). We classify the measures that can arise in terms of algebraic groups and homogeneous dynamics. Using the classification, we prove analogues of results of Siegel, Weil and Rogers about a Siegel summation formula and identities and bounds involving higher moments. We deduce results about asymptotics, with error estimates, of point-counting and patch-counting for typical cut-and-project se…

Model sets with positive entropy in Euclidean cut and project schemes

We construct model sets arising from cut and project schemes in Euclidean spaces whose associated Delone dynamical systems have positive toplogical entropy. The construction works both with windows that are proper and with windows that have empty interior. In a probabilistic construction, the entropy almost surely turns out to be proportional to the measure of the boundary of the window.

具有多面体窗口的切割投影集合 I：复杂度

Cut and project sets with polytopal window I: complexity

本文计算了多面体切割投影集合的复杂度函数增长率，提出拟典范条件以正确关联接受域与切割区域，并纠正了文献中关于几乎典范条件的错误。

切割与投影集的尖锐密度偏差：一种通过格点计数的方法

Sharp density discrepancy for cut and project sets: An approach via lattice point counting

本文研究切割与投影集的点计数渐近波动，获得了依赖于格丢番图性质的偏差一致上下界，并证明了其尖锐性，同时找到了与格双Lipschitz等价的最大切割与投影集类。

通过切割和投影方案构造Toeplitz数组

Constructing Toeplitz arrays via cut and project schemes

本文通过切割和投影方案，构造了与残余有限拓扑群相关的Toeplitz数组，并展示了其与模型集的一一对应关系，同时提出了具有特定遍历测度性质的不规则Toeplitz数组的例子。

具有一般多面体窗口的割投影集合的线性重复性刻画

A characterisation of linear repetitivity for cut and project sets with general polytopal windows

本文刻画了割投影集合的线性重复性，将其等价于低复杂度与丢番图条件，并推广至非凸、非齐次及带标记情形，覆盖广义彭罗斯铺砌。

Pisot substitution sequences, one dimensional cut-and-project sets and bounded remainder sets with fractal boundary

This paper uses a connection between bounded remainder sets in $\mathbb{R}^d$ and cut-and-project sets in $\mathbb{R}$ together with the fact that each one-dimensional Pisot substitution sequence is bounded distance equivalent to some lattice in order to construct several bounded remainder sets with fractal boundary. Moreover it is shown that there are cut-and-project sets being not bounded distance equivalent to each other even if they are locally indistinguishable, more precisely: even if they are contained in the same hull.

Complexity of Cut-and-Project Sets of Polytopal Type in Special Homogeneous Lie Groups

The aim of this paper is to determine the asymptotic growth rate of the complexity function of cut-and-project sets in the non-abelian case. In the case of model sets of polytopal type in homogeneous two-step nilpotent Lie groups we can establish that the complexity function asymptotically behaves like $r^{homdim(G) dim(H)}$. Further we generalize the concept of acceptance domains to locally compact second countable groups.

Pure point diffraction and cut and project schemes for measures: The smooth case

We present cut and project formalism based on measures and continuous weight functions of sufficiently fast decay. The emerging measures are strongly almost periodic. The corresponding dynamical systems are compact groups and homomorphic images of the underlying torus. In particular, they are strictly ergodic with pure point spectrum and continuous eigenfunctions. Their diffraction can be calculated explicitly. Our results cover and extend corresponding earlier results on dense Dirac combs and continuous weight functions with compact support. They also mark a clear difference in terms of facto…

在庞加莱盘上研究切和投影方案：从紧致富士扬群到混沌德洛内集

Cut and project schemes in the Poincaré disc: From cocompact Fuchsian groups to chaotic Delone sets

本文研究了基于紧致富士扬群的切和投影方案，证明了其生成的德洛内集为混沌结构，并扩展了相关研究。

有界余集、有界距离等价的切割投影集与等可分解性

Bounded remainder sets, bounded distance equivalent cut-and-project sets, and equidecomposability

利用Cieśla和Sabok的可测Hall定理，证明了相同测度的有界余集在完全无理向量平移下可测等可分解，以及有界距离等价的切割投影集对应的窗口可测等可分解；对一维情况，多面体窗口可得到多面体分解，但高维不成立。

Pure Discrete Spectrum and Regular Model Sets in Unimodular Substitution Tilings on R^d

We consider primitive substitution tilings on R^d whose expansion maps are unimodular. We assume that all the eigenvalues of the expansion maps are algebraic conjugates with the same multiplicity. In this case, we can construct a cut-and-project scheme with a Euclidean internal space. Under some additional condition, we show that if the substitution tiling has pure discrete spectrum, then the corresponding representative point sets are regular model sets in that cut-and-project scheme.

Perfectly ordered quasicrystals and the Littlewood conjecture

Linearly repetitive cut and project sets are mathematical models for perfectly ordered quasicrystals. In a previous paper we presented a characterization of linearly repetitive cut and project sets. In this paper we extend the classical definition of linear repetitivity to try to discover whether or not there is a natural class of cut and project sets which are models for quasicrystals which are better than `perfectly ordered'. In the positive direction, we demonstrate an uncountable collection of such sets (in fact, a collection with large Hausdorff dimension) for every choice of dimension of…