Let $x\in \mathbb T^d=\mathbb R^d/\mathbb Z^d$ and $p/q\in\mathbb Q^d/\mathbb Z^d$ be such that $\|x-p/q\| \leqslant t$ (with $t$ small) and $q\leqslant Q$ can I say that for all $m\in\mathbb N^\star$ (large), I can find $p'\in \mathbb Z^d$ and $1\leqslant q'\leqslant Qm$ such that $\|x- p'/q'\|\leqslant t/m $ (or even $\|x-p'/q'\|\leqslant C \frac t{m^\alpha}$ for some $\alpha>0$ and $C$ depending only on $d$)
remark : what I call $\|\;.\;\|$ on $\mathbb T^d$ is an abuse of notations, it is the usual distance on the torus : $\|x\|=\inf_{p\in\mathbb Z^d} \|x+p\|$.
My first idea was to decompose $[p/q-t,p/q+t]^d$ into boxes of length $t/m$ each one of them countaining a rational number of the form $p'/(Qm)$ but this does not work if $t\leqslant \frac 1 Q$. An other idea would be to use the pigeon hole principle but I can not write it.
Thanks in advance !