On the Duffin-Schaeffer conjecture

Abstract

Let $ψ:\mathbb{N}\to\mathbb{R}_{\ge0}$ be an arbitrary function from the positive integers to the non-negative reals. Consider the set $\mathcal{A}$ of real numbers $α$ for which there are infinitely many reduced fractions $a/q$ such that $|α-a/q|\le ψ(q)/q$. If $\sum_{q=1}^\infty ψ(q)φ(q)/q=\infty$, 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 $|α- a/q|\le ψ(q)/q$, giving a refinement of Khinchin's Theorem.