Extremal properties of product sets

We find the nearly optimal size of a set $A\subset [N] := \{1,\dots ,N\}$ so that the product set $AA$ satisfies either (i) $|AA| \sim |A|^2/2$ or (ii) $|AA| \sim |[N][N]|$. This settles problems recently posed in a paper of J. Cilleruelo, D. S. Ramana and O. Ramaré.