Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices

Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices, II

Recent results of several authors have led to constructions of parallelotopes which are bounded remainder sets for totally irrational toral rotations. In this brief note we explain, in retrospect, how some of these results can easily be obtained from a geometric argument which was previously employed by Duneau and Oguey in the study of deformation properties of mathematical models for quasicrystals.

Sets of bounded discrepancy for multi-dimensional irrational rotation

We study bounded remainder sets with respect to an irrational rotation of the $d$-dimensional torus. The subject goes back to Hecke, Ostrowski and Kesten who characterized the intervals with bounded remainder in dimension one. First we extend to several dimensions the Hecke-Ostrowski result by constructing a class of $d$-dimensional parallelepipeds of bounded remainder. Then we characterize the Riemann measurable bounded remainder sets in terms of "equidecomposability" to such a parallelepiped. By constructing invariants with respect to this equidecomposition, we derive explicit conditions for…

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.

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。

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。