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

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.