Extremal problems for GCDs

Abstract

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

One remark added