PROVED
This has been solved in the affirmative.
For any function $f:\mathbb{N}\to \mathbb{N}$ the property that, for almost all $\alpha$\[\left\lvert \alpha-\frac{p}{q}\right\rvert < \frac{f(q)}{q}\]has infinitely many solutions with $(p,q)=1$, is equivalent to\[\sum_{q\geq 1}\phi(q)\frac{f(q)}{q}=\infty.\]
The
Duffin-Schaeffer conjecture. It is easy to prove that the latter follows from the former. Erdős proved this in the special case when $f(q)q$ is bounded.
The full conjecture was proved by Koukoulopoulos and Maynard
[KoMa20].
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When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #999, https://www.erdosproblems.com/999, accessed 2026-01-16
