A note on product sets of random sets

Given two sets of positive integers A and B, let AB: = { ab: a∈ A, b∈ B} be their product set and put Ak: = A⋯ A (k times A) for any positive integer k. Moreover, for every positive integer n and every α= α(n) ∈ [0 , 1] , let B(n, α) denote the probabilistic model in which a random set A⊆ { 1 , … , n} is constructed by choosing independently every element of { 1 , … , n} with probability α. We prove that if A1, … , As are random sets in B(n1, α1) , … , B(ns, αs) , respectively, k1, … , ks are fixed positive integers, αini→ + ∞, and 1 / αi does not grow too fast in terms of a product of log nj; then |A1k1⋯Asks|∼|A1|k1k1!⋯|As|ksks! with probability 1 - o(1). This is a generalization of a result of Cilleruelo, Ramana, and Ramaré [3], who considered the case s= 1 and k1= 2.