On the Duffin-Schaeffer conjecture

Abstract

Let $\psi :\mathbb {N}\to \mathbb {R}_{\geqslant 0}$ 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 $|\alpha -a/q|\leqslant \psi (q)/q$. 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 $|\alpha – a/q|\leqslant \psi (q)/q$, giving a refinement of Khinchin’s Theorem.