Extremal properties of product sets

Abstract

We find nearly the optimal size of a set $A\subset [N]:=\{1,...,N\}$ so that the product set $AA$ satisfies either (i) $|AA|\sim |A{|}^2/2$ or (ii) $|AA|\sim |[N][N]|$. This settles problems raised in a recent article of Cilleruelo, Ramana and Ramare.

v2. minor corrections