On maximal product sets of random sets

For every positive integer N and every $\alpha \in [0,1)$, let $B(N,\alpha )$ denote the probabilistic model in which a random set $A\subset \{1,\dots ,N\}$ is constructed by choosing independently every element of $\{1,\dots ,N\}$ with probability $\alpha$. We prove that, as $N⟶+\mathrm{\infty }$, for every $A$ in $B(N,\alpha )$ we have $\mathrm{\mid }AA\mathrm{\mid }\sim \mathrm{\mid }A{\mathrm{\mid }}^2\mathrm{/}2$ with probability $1-o(1)$, if and only if $$\frac{\log ({\alpha }^2(\log N{)}^{\log 4-1})}{\sqrt{\log \log N}}⟶-\mathrm{\infty }\mathrm{.}$$ This improves a theorem of Cilleruelo, Ramana and Ramar'e, who proved the above asymptotic between $\mathrm{\mid }AA\mathrm{\mid }$ and $\mathrm{\mid }A{\mathrm{\mid }}^2\mathrm{/}2$ when $\alpha =o(1\mathrm{/}\sqrt{\log N})$, and supplies a complete characterization of maximal product sets of random sets.