Let $\psi:N\toR_\ge0$ be an arbitrary function from the
positive integers to the non-negative reals. Consider the set $A$ of
real numbers $\alpha$ for which there are infinitely many reduced fractions
$a/q$ such that $|\alpha-a/q|\psi(q)/q$. If $\sum_q=1^ınfty
\psi(q)\phi(q)/q=ınfty$, we show that $A$ 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 $|- a/q|\psi(q)/q$, giving a refinement of Khinchin's
Theorem.
%0 Journal Article
%1 koukoulopoulos2019duffinschaeffer
%A Koukoulopoulos, Dimitris
%A Maynard, James
%D 2019
%K mathematics theory
%T On the Duffin-Schaeffer conjecture
%U http://arxiv.org/abs/1907.04593
%X Let $\psi:N\toR_\ge0$ be an arbitrary function from the
positive integers to the non-negative reals. Consider the set $A$ of
real numbers $\alpha$ for which there are infinitely many reduced fractions
$a/q$ such that $|\alpha-a/q|\psi(q)/q$. If $\sum_q=1^ınfty
\psi(q)\phi(q)/q=ınfty$, we show that $A$ 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 $|- a/q|\psi(q)/q$, giving a refinement of Khinchin's
Theorem.
@article{koukoulopoulos2019duffinschaeffer,
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.},
added-at = {2019-07-11T18:39:28.000+0200},
author = {Koukoulopoulos, Dimitris and Maynard, James},
biburl = {https://www.bibsonomy.org/bibtex/2b750426cf32dcf7bf1d200eca787b555/kirk86},
description = {[1907.04593] On the Duffin-Schaeffer conjecture},
interhash = {b416ec1d1cf970b4fb4ef3b9dcdde096},
intrahash = {b750426cf32dcf7bf1d200eca787b555},
keywords = {mathematics theory},
note = {cite arxiv:1907.04593Comment: 45 pages},
timestamp = {2019-07-11T18:39:28.000+0200},
title = {On the Duffin-Schaeffer conjecture},
url = {http://arxiv.org/abs/1907.04593},
year = 2019
}