On maximal product sets of random sets

Abstract

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 $N⟶+\infty$, for every $A$ in $B(N,\alpha )$ we have $|AA|\sim |A{|}^2/2$ with probability $1-o(1)$, if and only if $$\frac{\log ({\alpha }^2(\log N{)}^{\log 4-1})}{\sqrt{\log \log N}}⟶-\infty .$$ This improves a theorem of Cilleruelo, Ramana and Ramaré, 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.

arXiv commentFixed some typos