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July 2020 On the Duffin-Schaeffer conjecture
Dimitris Koukoulopoulos, James Maynard
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Ann. of Math. (2) 192(1): 251-307 (July 2020). DOI: 10.4007/annals.2020.192.1.5

Abstract

Let $\psi: \mathbb{N}\to \mathbb{R}_{\ge 0}$ 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 $\sum_{q=1}^\infty \psi(q)\varphi(q)/q = \infty$, 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.

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Dimitris Koukoulopoulos. James Maynard. "On the Duffin-Schaeffer conjecture." Ann. of Math. (2) 192 (1) 251 - 307, July 2020. https://doi.org/10.4007/annals.2020.192.1.5

Information

Published: July 2020
First available in Project Euclid: 21 December 2021

Digital Object Identifier: 10.4007/annals.2020.192.1.5

Subjects:
Primary: 11J83
Secondary: 05C40

Keywords: compression arguments , density increment , diophantine approximation , Duffin-Schaeffer conjecture , graph theory , metric number theory

Rights: Copyright © 2020 Department of Mathematics, Princeton University

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Vol.192 • No. 1 • July 2020
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