Erdős's integer dilation approximation problem and GCD graphs

Abstract

Let $\mathcal{A}\subset\mathbb{R}_{\geqslant1}$ be a countable set such that $\limsup_{x\to\infty}\frac{1}{\log x}\sum_{α\in\mathcal{A}\cap[1,x]}\frac{1}α>0$. We prove that, for every $\varepsilon>0$, there exist infinitely many pairs $(α, β)\in \mathcal{A}^2$ such that $α\neq β$ and $|nα-β| <\varepsilon$ for some positive integer $n$. This resolves a problem of Erdős 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.