General Events - Department of Mathematics

Speaker: Dimitris Koukoulopoulos

• Title: Erdős's integer dilation approximation problem 

Abstract: Let 𝒜 ⊂ ℝ ≥ 1 be a countable set such that limsup x→∞(1/log x)Σ α∈𝒜∩ [1,x](1/α)>0. Erdős conjectured in 1948 that, for every ε>0, there exist infinitely many pairs (α, β)∈𝒜² such that α ≠ β and |nα -β| <ε for some positive integer n. When 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 𝒜 contains "enough elements" all of whose 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.