Fix $\alpha \in \mathbf{R}$. The classical Dirichlet's approximation theorem states there exist infinitely many rationals $p/q$ such that $$ \left|\alpha-\frac{p}{q}\right|<\frac{1}{q^2}. $$

Question.Fix $\alpha \in \mathbf{R}$. Is it true that there exist infinitely many rationals $p/q$ such that $$ 0\le \alpha- \frac{p}{q}\ll\frac{1}{q^2}\,\,? $$