The Duffin-Schaeffer conjecture with extra divergence

Abstract

The Duffin-Schaeffer conjecture is a fundamental unsolved problem in metric number theory. It asserts that for every non-negative function $ψ:~\mathbb{N} \rightarrow \mathbb{R}$ for almost all reals $x$ there are infinitely many coprime solutions $(a,n)$ to the inequality $|nx - a| < ψ(n)$, provided that the series $\sum_{n=1}^\infty ψ(n) φ(n) /n$ is divergent. In the present paper we prove that the conjecture is true under the "extra divergence" assumption that divergence of the series still holds when $ψ(n)$ is replaced by $ψ(n) / (\log n)^\varepsilon$ for some $\varepsilon > 0$. This improves a result of Beresnevich, Harman, Haynes and Velani, and solves a problem posed by Haynes, Pollington and Velani.