A note on product sets of random sets

TL;DR

This paper investigates the asymptotic size of product sets formed from multiple random subsets of integers, extending previous results to more complex multi-set products under probabilistic models.

Contribution

It generalizes earlier work by analyzing the product sets of multiple random sets with various exponents, providing asymptotic formulas under broad probabilistic conditions.

Findings

Product set sizes approximate the product of individual set sizes raised to powers divided by factorials.

Results hold with high probability as set sizes grow large.

Extends prior single-set results to multiple sets and exponents.

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 Ramar\'e, who considered the case $s = 1$ and $k_1 = 2$.