The Duffin-Schaeffer Conjecture with extra divergence

TL;DR

This paper proves that the set of real numbers approximable by rationals with a certain divergence condition has full measure or dimension, advancing the understanding of the Duffin-Schaeffer Conjecture in metric number theory.

Contribution

It establishes new divergence criteria ensuring full measure and dimension for approximation sets, providing partial progress on the Duffin-Schaeffer Conjecture.

Findings

Full Lebesgue measure when a weighted sum diverges with epsilon > 0

Full Hausdorff dimension when the sum diverges with epsilon = 0

Progress towards the Duffin-Schaeffer Conjecture in metric number theory

Abstract

Given a nonnegative function $\psi : \N \to \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) $. A consequence of our main result is that $W(\psi)$ is of full Lebesgue measure if there exists an $\epsilon > 0 $ such that $$ \textstyle \sum_{n\in\N}(\frac{\psi(n)}{n})^{1+\epsilon}\varphi (n)=\infty . $$ The Duffin-Schaeffer Conjecture is the corresponding statement with $\epsilon = 0$ and represents a fundamental unsolved problem in metric number theory. Another consequence is that $W(\psi)$ is of full Hausdorff dimension if the above sum with $\epsilon = 0$ diverges; i.e. the dimension analogue of the Duffin-Schaeffer Conjecture is true.