Extremal problems for GCDs

Source abstract

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