%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 }