Abstract
Let $\psi: \mathbb{N}\to \mathbb{R}_{\ge 0}$ be an arbitrary function from the positive integers to the non-negative reals. Consider the set $\mathcal{A}$ of real numbers $\alpha$ for which there are infinitely many reduced fractions $a/q$ such that $|\alpha-a/q| \le \psi(q)/q$. If $\sum_{q=1}^\infty \psi(q)\varphi(q)/q = \infty$, we show that $\mathcal{A}$ has full Lebesgue measure. This answers a question of Duffin and Schaeffer. As a corollary, we also establish a conjecture due to Catlin regarding non-reduced solutions to the inequality $|\alpha-a/q| \le \psi(q)/q)$, giving a refinement of Khinchin's Theorem.
Citation
Dimitris Koukoulopoulos. James Maynard. "On the Duffin-Schaeffer conjecture." Ann. of Math. (2) 192 (1) 251 - 307, July 2020. https://doi.org/10.4007/annals.2020.192.1.5
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