Exchanged toric developments and bounded remainder sets
type: Item
instance of: scholarly article
title: Exchanged toric developments and bounded remainder sets
zbMATH Open document ID: 1272.11087
author: V. G. Zhuravlev
published in: Journal of Mathematical Sciences (New York)
publication date: 2012-11-29T00:00:00Z
review text: The paper concerns the distribution of fractional parts of integer multiples of vectors in \(\mathbb{R}^D\). Depending on a vector \(s \in \mathbb{R}^D\), the author defines polyhedra \(F^s \subseteq \mathbb{R}^D\), so-called stretched or mirror-stretched cubes, that give rise to a tiling \(\{F^s+l: l \in L\}\) of \(\mathbb{R}^D\) based on a lattice \(L=L(s) \subseteq \mathbb{R}^D\). Another parameter \(0 < \lambda < 1\) is used for introducing a particular congruence by dissection of \(F^s\) with itself that is realized by a dissection of \(F^s\) into polyhedra \(F^{s,\lambda}_k\), \(k=0,\ldots,D\). The map \(x \mapsto x\, \mathrm{mod}\, L\) of \(F^s\) onto the torus \(\mathbb{T}^s = \mathbb{R}^D/L\) induces a related dissection of \(\mathbb{T}^s\) into \(\mathbb{T}^{s,\lambda}_k\), \(k=0,\ldots,D\). If the parameters \(s\) and \(\lambda\) satisfy a condition of irrationality, the sets \(\mathbb{T}^{s,\lambda}_k\) show the behavior of bounded remainder sets: for every \(n \in \{1,2,\ldots\}\) and every \(l \in L\), the shift vector \(\beta=\frac{\lambda s + l}{n}\) satisfies \[ \left|\{j: j \beta \,\mathrm{mod}\,L \in \mathbb{T}^{s,\lambda}_k, 0 \leq j < i\}| - i \frac{\mathrm{vol}(\mathbb{T}^{s,\lambda}_k)}{\mathrm{vol}(\mathbb{T}^s)} \right| \leq c(s,k) n \] for \(i=0,1,2,\ldots\) Similar consequences are obtained for certain kinds of dissections of the above tilings of tori.
reviewed by: Christian Richter
Mathematics Subject Classification ID: 11J71
Mathematics Subject Classification ID: 52B45
Mathematics Subject Classification ID: 52C07
Mathematics Subject Classification ID: 52C22
zbMATH DE Number: 6111011
zbMATH Keywords: fractional part with respect to a lattice
zbMATH Keywords: distribution of fractional parts
zbMATH Keywords: bounded remainder set
zbMATH Keywords: dissection of the torus
zbMATH Keywords: tiling
zbMATH Keywords: congruence by dissection
MaRDI profile type: Publication
full work available at URL: s10958-012-0894-0
OpenAlex ID: W2076955127
cites work: Free and nonfree Voronoi polyhedra
cites work: Über die Verteilung der Vielfachen einer komplexen Zahl nach dem Modul des Einheitsquadrats
DOI: 10.1007/S10958-012-0894-0
description: scientific article; zbMATH DE number 6111011