Number Theory Web Seminar

Dimitris Koukoulopoulos, Erdős's integer dilation approximation problem
(University of Montreal)

Thursday, April 17, 2025 (8am PDT, 11am EDT, 4pm BST, 5pm CEST, 6pm IDT, 8:30pm IST, 11pm CST)
Friday, April 18, 2025 (1am AEST, 3am NZST)

Abstract: Let $\mathcal{A}\subset\mathbb{R}_{\geqslant1}$ be a countable set such that $\limsup_{x\to\infty}\frac{1}{\log x}\sum_{\alpha\in\mathcal{A}\cap[1,x]}\frac{1}{\alpha}>0$. Erdős conjectured in 1948 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 $n$. When $\mathcal{A}$ is a set of integers, the conjecture follows by work of Erdős and Behrend on primitive sets of integers from the 1930s. Moreover, if $\mathcal{A}$ contains ``enough elements" all of pairwise ratios are irrational, then Haight proved Erdős's conjecture in 1988. In this talk, I will present recent joint work with Youness Lamzouri and Jared Duker Lichtman that solves the conjecture in full generality. A critical role in the proof is played by the machinery of GCD graphs, which were introduced by Koukoulopoulos-Maynard in the proof of the Duffin--Schaeffer conjecture in Diophantine approximation.