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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 $$ 0\le \alpha- \frac{p}{q}\ll\frac{1}{q^2}\,\,? $$

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  • $\begingroup$ What does the $\ll$ mean in this context? $\endgroup$
    – Vincent
    Commented Jan 22, 2020 at 12:18
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    $\begingroup$ "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 $p/q$" $\endgroup$
    – Paolo Leonetti
    Commented Jan 22, 2020 at 12:20

1 Answer 1

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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

$$ 0\leq \alpha - \frac{p_n}{q_n} \leq \frac{1}{q_n^2}.$$

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