A note on the Duffin-Schaeffer conjecture with slow divergence

TL;DR

This paper proves that the set of real numbers satisfying a certain Diophantine approximation condition has full measure under a slow divergence criterion, extending previous results that required faster divergence conditions.

Contribution

It establishes a new slow divergence criterion ensuring the full measure of the set related to the Duffin-Schaeffer conjecture, broadening the understanding of divergence conditions in metric number theory.

Findings

Proves full measure of $W(\psi)$ under slow divergence condition.

Extends previous results by relaxing divergence rate requirements.

Provides a new criterion involving logarithmic growth conditions.

Abstract

For a non-negative function $\psi: ~ \N \mapsto \R$, let $W(\psi)$ denote the set of real numbers $x$ for which the inequality $|n x - a| < \psi(n)$ has infinitely many coprime solutions $(a,n)$. The Duffin--Schaeffer conjecture, one of the most important unsolved problems in metric number theory, asserts that $W(\psi)$ has full measure provided {equation} \label{dsccond} \sum_{n=1}^\infty \frac{\psi(n) \varphi(n)}{n} = \infty. {equation} Recently Beresnevich, Harman, Haynes and Velani proved that $W(\psi)$ has full measure under the \emph{extra divergence} condition $$ \sum_{n=1}^\infty \frac{\psi(n) \varphi(n)}{n \exp(c (\log \log n) (\log \log \log n))} = \infty \qquad \textrm{for some $c>0$}. $$ In the present note we establish a \emph{slow divergence} counterpart of their result: $W(\psi)$ has full measure, provided\eqref{dsccond} holds and additionally there exists some $c>0$ such that $$ \sum_{n=2^{2^h}+1}^{2^{2^{h+1}}} \frac{\psi(n) \varphi(n)}{n} \leq \frac{c}{h} \qquad \textrm{for all \quad $h \geq 1$.} $$