On maximal product sets of random sets

For every positive integer N and α∈[0,1), let B(N,α) denote the probabilistic model in which a random set A⊂{1,…,N} is constructed by choosing independently element of {1,…,N} with probability α. We prove that, as N⟶+∞, for A we have |AA|∼|A|2/2 1−o(1), if only iflog⁡(α2(log⁡N)log⁡4−1)log⁡log⁡N⟶−∞. This improves on theorem Cilleruelo, Ramana Ramaré, who proved above asymptotic between |AA| |A|2/2 when α=o(1/log⁡N), supplies complete characterization maximal product sets sets.