A note on the Duffin-Schaeffer conjecture

Abstract

Given a sequence of real numbers $\{\psi (n){\}}_{n\in \mathbb{N}}$ with $0\le \psi (n)<1$, let $W(\psi )$ denote the set of $x\in [0,1]$ for which $|xn-m|<\psi (n)$ for infinitely many coprime pairs $(n,m)\in \mathbb{N}\times \mathbb{Z}$. The purpose of this note is to show that if there exists an $\epsilon >0$ such that ${\sum }_{n\in \mathbb{N}}\psi (n{)}^{1+\epsilon }\cdot \frac{\phi (n)}{n}=\infty ,$ then the Lebesgue measure of $W(\psi )$ equals 1.

Accepted by the Uniform Distribution Theory journal