Fix $\alpha \in \mathbf{R}$. The classical [Dirichlet's approximation theorem](https://en.wikipedia.org/wiki/Dirichlet%27s_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
>  $$
> 0\le \alpha- \frac{p}{q}\ll\frac{1}{q^2}\,\,?
> $$
> 
> 
> 



Sign in Dirichlet's approximation theorem
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


$$ 0\leq \alpha - \frac{p\_n}{q\_n} \leq \frac{1}{q\_n^2}.$$


