On the DuffinSchaeffer conjecture
Abstract
Let $\psi:\mathbb{N}\to\mathbb{R}_{\ge0}$ be an arbitrary function from the positive integers to the nonnegative reals. Consider the set $\mathcal{A}$ of real numbers $\alpha$ for which there are infinitely many reduced fractions $a/q$ such that $\alphaa/q\le \psi(q)/q$. If $\sum_{q=1}^\infty \psi(q)\phi(q)/q=\infty$, 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 nonreduced solutions to the inequality $\alpha  a/q\le \psi(q)/q$, giving a refinement of Khinchin's Theorem.
 Publication:

arXiv eprints
 Pub Date:
 July 2019
 arXiv:
 arXiv:1907.04593
 Bibcode:
 2019arXiv190704593K
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Combinatorics;
 11J83 (Primary);
 05C40 (Secondary)
 EPrint:
 Final version, 46 pages, to appear in Annals of Mathematics. Fixed a typo in equation (14.1) from the previous version