A note on the Duffin-Schaeffer conjecture

TL;DR

This paper proves that under certain divergence conditions involving the sum of b4(n)^{1+\u03b5} 7 (n)/n, the set of real numbers approximable infinitely often by coprime pairs has full Lebesgue measure.

Contribution

It establishes a new divergence criterion ensuring the Lebesgue measure of the set W(b4) is 1, advancing understanding of the Duffin-Schaeffer conjecture.

Findings

If b4(n)^{1+5} 7 (n)/n diverges, then measure of W(b4) is 1.

The result provides a partial solution to the Duffin-Schaeffer conjecture.

The proof links divergence of a weighted sum to measure-theoretic properties.

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.