Extremal problems for GCDs

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 $\delta \mathrm{\mid }A\mathrm{\mid }\mathrm{\mid }B\mathrm{\mid }$ pairs $(a,b) \in A \times B$then$|A||B| \ll_{\varepsilon} \delta^{-2 - \varepsilon} XY/D^2$\mathrm{.}Thisisanewresultevenwhen$\delta = 1$. The proof uses ideas of Koukoulopoulos and Maynard and some additional combinatorial arguments.