The Duffin–Schaeffer conjecture with extra divergence II

Abstract

In 2012 the authors set out a programme to prove the Duffin–Schaeffer conjecture for measures arbitrarily close to Lebesgue measure. In this paper we take a new step in this direction. Given a non-negative function $\psi : \mathbb N \rightarrow \mathbb R$, let $W(\psi )$ denote the set of real numbers $x$ such that $|nx -a| < \psi (n)$ for infinitely many reduced rationals $a/n \ (n>0)$. Our main result is that $W(\psi )$ is of full Lebesgue measure if there exists a $c > 0$ such that

$$\begin{aligned} \sum _{n\ge 16} \, \frac{\varphi (n) \psi (n)}{n \exp (c(\log \log n)(\log \log \log n))} \, = \, \infty \, . \end{aligned}$$