Erdős's integer dilation approximation problem and GCD graphs

Abstract

Let $\mathcal{A}\subset {\mathbb{R}}_{⩾1}$ be a countable set such that ${\mathrm{limsup}}_{x\to \infty }\frac{1}{\log x}{\sum }_{\hat{I}\pm \in \mathcal{A}\cap [1,x]}\frac{1}{\hat{I}}\pm >0$. We prove that, for every $\epsilon >0$, there exist infinitely many pairs $(\hat{I}\pm ,{\hat{I}}^2)\in {\mathcal{A}}^2$ such that $\hat{I}\pm \ne {\hat{I}}^2$ and $|n\hat{I}\pm -{\hat{I}}^2|<\epsilon$ 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.

47 pages, 1 figure