Extremal problems for GCDs

TL;DR

This paper establishes an upper bound on the product of sizes of two integer sets with a high proportion of pairs having a large gcd, extending previous results even in the case of complete pairwise gcd conditions.

Contribution

It introduces a new bound on the sizes of sets with many pairs having gcd at least D, using combinatorial and number-theoretic techniques, improving upon prior results.

Findings

Derived an upper bound for |A||B| in terms of δ, X, Y, and D

Extended the understanding of gcd distribution in large integer sets

Applied combinatorial methods to number theory problems

Abstract

We prove that if $A \subseteq [X, 2X]$ and $B \subseteq [Y, 2Y]$ are sets of integers such that $\gcd(a,b) \geq D$ for at least $\delta |A||B|$ pairs $(a,b) \in A \times B$ then $|A||B| \ll_{\varepsilon} \delta^{-2 - \varepsilon} XY/D^2$. This is a new result even when $\delta = 1$. The proof uses ideas of Koukoulopoulos and Maynard and some additional combinatorial arguments.