A three gap theorem for the adeles

TL;DR

This paper generalizes the classical three gap theorem to rotations on adelic tori, extending the understanding of gaps in Diophantine approximation through a lattice-based approach.

Contribution

It introduces a natural generalization of the three gap theorem for adelic tori, adapting lattice methods to this new setting.

Findings

Proves a three gap theorem for adelic tori.

Extends lattice-based gap analysis to adelic settings.

Provides a new perspective on Diophantine approximation gaps.

Abstract

We prove a natural generalization of the classical three gap theorem, for rotations on adelic tori. Our proof is an adaptation to the adeles of the lattice based approach to gaps problems in Diophantine approximation originally introduced by Marklof and Str\"ombergsson.