A note on product sets of random sets

Abstract

Given two sets of positive integers $A$ and $B$, let $AB:=\{ab:a\in A,b\in B\}$ be their product set and put $A^k:=A\cdots A$ ($k$ times $A$) for any positive integer $k$. Moreover, for every positive integer $n$ and every $\alpha \in [0,1]$, let $\mathcal{B}(n,\alpha )$ denote the probabilistic model in which a random set $A\subseteq \{1,\dots ,n\}$ is constructed by choosing independently every element of $\{1,\dots ,n\}$ with probability $\alpha$. We prove that if $A_1,\dots ,A_s$ are random sets in $\mathcal{B}(n_1,{\alpha }_1),\dots ,\mathcal{B}(n_s,{\alpha }_s)$, respectively, $k_1,\dots ,k_s$ are fixed positive integers, ${\alpha }_i{n}_i\to +\infty$, and $1/{\alpha }_i$ does not grow too fast in terms of a product of $\log n_j$; then $|A_1^{k_1}\cdots A_s^{k_s}|\sim \frac{|A_1{|}^{k_1}}{k_1!}\cdots \frac{|A_s{|}^{k_s}}{k_s!}$ with probability $1-o(1)$. This is a generalization of a result of Cilleruelo, Ramana, and Ramaré, who considered the case $s=1$ and $k_1=2$.

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