Une propriété de transfert en approximation diophantienne
Résumé
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 $x\mapsto x+\omega$ 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 precidely, 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.
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