On the Duffin-Schaeffer conjecture

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.

Final version, 46 pages, to appear in Annals of Mathematics. Fixed a typo in equation (14.1) from the previous version