Une propriété de transfert en approximation diophantienne

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

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