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 $$0\le \alpha- \frac{p}{q}\ll\frac{1}{q^2}\,\,?$$
• What does the $\ll$ mean in this context? – Vincent Jan 22 at 12:18
• "There exists an absolute constant $c>0$ such that the inequality $0\le \alpha-p/q \le c/q^2$ holds for infinitely many rationals $p/q$" – Paolo Leonetti Jan 22 at 12:20
Yes, this follows from considering the continued fraction of $$\alpha$$. If $$p_n/q_n$$ is the $$n$$th convergent to $$\alpha$$ and $$n$$ is odd then
$$0\leq \alpha - \frac{p_n}{q_n} \leq \frac{1}{q_n^2}.$$