Extremal properties of product sets

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