On the Duffin-Schaeffer conjecture

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 $a\mathrm{/}q$ such that $\mathrm{\mid }\alpha -a\mathrm{/}q\mathrm{\mid }\le \psi (q)\mathrm{/}q$. If $\sum_{q=1}^\infty \psi(q)\phi(q)/q=\infty$,weshowthat$$\mathcal{A}$$ 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 $\mathrm{\mid }\alpha -a\mathrm{/}q\mathrm{\mid }\le \psi (q)\mathrm{/}q$, giving a refinement of Khinchin’s Theorem.