Une propri\'et\'e de transfert en approximation diophantienne

TL;DR

This paper explores the relationship between the Diophantine properties of a vector on the torus and the growth rate of its periods, establishing conditions under which the vector is Diophantine.

Contribution

It demonstrates that certain growth conditions on the sequence of periods imply the vector's Diophantine nature, linking dynamical behavior to number-theoretic properties.

Findings

Sequences with controlled growth imply Diophantine vectors.

Non-resonant vectors with specific period bounds are Diophantine.

Provides a transfer property connecting periods and Diophantine conditions.

Abstract

Given a vector $\omega \in \mathbb{R}^n$,the sequence $T_i$ of periods is defined as the sequence of times of best returns near the origin of the translation $x \longmapsto x+\omega$ on the torus $\mathbb{T}^n$. In the present paper, we study how the Diophantine properties of $\omega$ can be expressed considering the sequence of its periods. More precisely, we prove that, if the vector $\omega$ is not resonant,and if the sequence of periods satisfy the inequality$T_{i+1} \leq CT_i^{1+\tau}$ with$\tau<(n-1)^{-1}$, then the vector $\omega$ is Diophantine.