  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.
