# On the Duffin-Schaeffer conjecture

## TL;DR

This paper proves that the set of real numbers approximable by fractions with a certain error rate has full measure under a divergence condition, resolving a longstanding conjecture and refining classical approximation theorems.

## Contribution

It establishes the Duffin-Schaeffer conjecture for Lebesgue measure and confirms Catlin's conjecture on non-reduced solutions, advancing metric number theory.

## Key findings

- Proves the Duffin-Schaeffer conjecture for full measure.
- Confirms Catlin's conjecture on non-reduced solutions.
- Refines Khinchin's Theorem with new approximation conditions.

## Abstract

Let $\psi:\mathbb{N}\to\mathbb{R}_{\ge0}$ be an arbitrary function from the positive integers to the non-negative reals. Consider the set $\mathcal{A}$ of real numbers $\alpha$ for which there are infinitely many reduced fractions $a/q$ such that $|\alpha-a/q|\le \psi(q)/q$. If $\sum_{q=1}^\infty \psi(q)\phi(q)/q=\infty$, we show that $\mathcal{A}$ has full Lebesgue measure. This answers a question of Duffin and Schaeffer. As a corollary, we also establish a conjecture due to Catlin regarding non-reduced solutions to the inequality $|\alpha - a/q|\le \psi(q)/q$, giving a refinement of Khinchin's Theorem.