A note on the Duffin-Schaeffer conjecture

Abstract

Given a sequence of real numbers $\{ψ(n)\}_{n\in\mathbb{N}}$ with $0\leq ψ(n)<1$, let $W(ψ)$ denote the set of $x\in[0,1]$ for which $|xn-m|<ψ(n)$ 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 $ε>0$ such that $\sum_{n\in\mathbb{N}}ψ(n)^{1+ε}\cdot\frac{φ(n)}{n}=\infty,$ then the Lebesgue measure of $W(ψ)$ equals 1.

Accepted by the Uniform Distribution Theory journal