On maximal product sets of random sets

For every positive integer Nand every alpha is an element of[0, 1), let B(N, alpha) denote the probabilistic model in which a random set A subset of {1, ... , N} is constructed by choosing independently every element of {1, ... , N} with probability alpha. We prove that, as N -> +infinity, for every Ain B(N, alpha) we have vertical bar AA vertical bar similar to vertical bar A vertical bar(2)/2 with probability 1 - o(1), if and only if log(alpha(2)(log N)(log 4-1))/root log log N -> -infinity. This improves on a theorem of Cilleruelo, Ramana and Ramare, who proved the above asymptotic between vertical bar AA vertical bar and vertical bar A vertical bar(2)/2when alpha = o(1/root log N), and supplies a complete characterization of maximal product sets of random sets. (c) 2021 Elsevier Inc. All rights reserved.