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 $N\longrightarrow +\infty$, for every $A$ in $B(N, \alpha)$ we have $|AA|\ \sim |A|^2/2$ with probability $1-o(1)$, if and only if $$\frac{\log(\alpha^2(\log N)^{\log 4-1})}{\sqrt{\log\log N}}\longrightarrow-\infty.$$ This improves a theorem of Cilleruelo, Ramana and Ramar\'e, who proved the above asymptotic between $|AA|$ and $|A|^2/2$ when $\alpha=o(1/\sqrt{\log N})$, and supplies a complete characterization of maximal product sets of random sets.