Let $\epsilon>0$ and $x_1,\ldots,x_n\in\mathbb R^n$ such that $x_1,\ldots,x_n$ are linearly independent over $\mathbb Q$.
We know from the pigeonhole principle that there exists infinitely many $q=(q_1,\ldots,q_n)\in\mathbb Z^n$ such that
$$\vert q_1x_1+\cdots+q_nx_n\vert \leqslant \Vert q\Vert^{-n\varepsilon}.$$
What can we say about the growth rate of the vectors $q$?
I am interested in any references or results regarding this growth rate, whether it is in a special case or in the general one.