Dirichlet's theorem on diophantine approximation states that given $\alpha\in\mathbb{R}$ and $Q>0$, one can always find a rational approximation $a/q$ with $1\leq q\leq Q$ such that $|\alpha - a/q|< 1/qQ$. I am curious if it is possible to take q to be large, more like of size $Q$. More concretely, here is my question:
Q. Given $\alpha\in\mathbb{R}$ and $\epsilon>0$, does there exist a $Q_0$ (depending on $\alpha$ and $\epsilon$) such that for every $Q>Q_0$, one can find integers $a$ and $q$ satisfying $Q^{1-\epsilon}<q<Q^{1+\epsilon}$ and $|\alpha - a/q|< 1/qQ$?