Erd\H{o}s's integer dilation approximation problem and GCD graphs

Abstract

Let $\mathcal{A}\subset\mathbb{R}_{\geqslant1}$ be a countable set such that $\limsup_{x\to\infty}\frac{1}{\log x}\sum_{\alpha\in\mathcal{A}\cap[1,x]}\frac{1}{\alpha}>0$. We prove that, for every $\varepsilon>0$, there exist infinitely many pairs $(\alpha, \beta)\in \mathcal{A}^2$ such that $\alpha\neq \beta$ and $|n\alpha-\beta| <\varepsilon$ for some positive integer $n$. This resolves a problem of Erd\H{o}s from 1948. A critical role in the proof is played by the machinery of GCD graphs, which were introduced by the first author and by James Maynard in their work on the Duffin--Schaeffer conjecture in Diophantine approximation.