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

TL;DR

This paper discusses geometric methods to construct bounded remainder sets and cut-and-project sets that are close to lattices, building on recent advances in toral rotations and quasicrystal models.

Contribution

It offers a geometric approach to derive results on bounded remainder sets and cut-and-project sets, connecting them to lattice approximations and quasicrystal deformation analysis.

Findings

Constructs bounded remainder sets using geometric arguments.

Links results on toral rotations to quasicrystal models.

Provides a unified geometric perspective on these sets.

Abstract

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.