Erdős's integer dilation approximation problem and GCD graphs

Let $\mathcal{A}\subset {\mathbb{R}}_{⩾1}$ be a countable set such that ${\mathrm{lim\hspace{0.17em}sup}}_{x\to \mathrm{\infty }}\frac{1}{\log x}{\sum }_{\alpha \in \mathcal{A}\cap [1,x]}\frac{1}{\alpha }>0$. We prove that, for every $\epsilon >0$, there exist infinitely many pairs $(\alpha ,\beta )\in {\mathcal{A}}^2$ such that $\alpha \mathrm{\ne }\beta$ and $\mathrm{\mid }n\alpha -\beta \mathrm{\mid }<\epsilon$ 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.