Bounded remainder sets for rotations on the adelic torus

TL;DR

This paper constructs bounded remainder sets of all volumes for irrational rotations on the adelic torus, combining dynamical systems, harmonic analysis, and geometric methods related to quasicrystals.

Contribution

It provides an explicit construction of bounded remainder sets for all volumes on the adelic torus, extending previous results to a new setting.

Findings

Explicit construction of bounded remainder sets for all volumes

Application of harmonic analysis and geometric methods

Advancement in understanding rotations on adelic structures

Abstract

In this paper we give an explicit construction of bounded remainder sets of all possible volumes, for any irrational rotation on the adelic torus $\mathbb A/\mathbb Q$. Our construction involves ideas from dynamical systems and harmonic analysis on the adeles, as well as a geometric argument which originated in the study of deformation properties of mathematical quasicrystals.