The Duffin-Schaeffer conjecture with extra divergence

TL;DR

This paper proves the Duffin-Schaeffer conjecture under an additional divergence condition, advancing understanding in metric number theory and solving a previously posed open problem.

Contribution

It establishes the conjecture's validity when the divergence persists after dividing the approximation function by a logarithmic factor.

Findings

Proves the Duffin-Schaeffer conjecture with extra divergence condition.

Improves upon previous results by relaxing divergence requirements.

Solves a problem posed by Haynes, Pollington, and Velani.

Abstract

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