Best Diophantine Approximations and Multidimensional Three Distance Theorem

TL;DR

This paper explores the connection between Diophantine approximation and the multidimensional Three Distance Theorem, deriving known results and presenting new findings, while also discussing inverse applications.

Contribution

It demonstrates how to derive multidimensional Three Distance Theorem results from Diophantine approximation and introduces new liminf version results, also exploring inverse applications.

Findings

Known results about multidimensional three distance theorem derived from Diophantine approximations

New liminf version results obtained

Discussion on inverse applications between the two topics

Abstract

In 1996 N. Chevallier proved a beautiful lemma which connects Diophantine approximation and multidimensional generalizations of the famous Three Distance Theorem. Using this lemma we show how known results about multidimensional three distance theorem can be deduced from certain known results dealing with the best Diophantine approximations. Also we obtain some new results about liminf version of the problem. Beside this, we discuss the inverse problem: how results about multidimensional three distance theorem can be applied to study best Diophantine approximations.