A note on the Duffin-Schaeffer conjecture

Abstract

Given a sequence of real numbers $\{\psi(n)\}_{n\in\mathbb{N}}$ with $0\leq \psi(n)<1$, 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 $\sum_{n\in\mathbb{N}}\psi(n)^{1+\epsilon}\cdot\frac{\varphi(n)}{n}=\infty,$ then the Lebesgue measure of $W(\psi)$ equals 1.