A note on the natural density of product sets

TL;DR

This paper investigates the natural density of product sets of natural numbers, proving that the product of two sets with density 1 also has density 1, and constructing sets with high density whose product set has low density.

Contribution

It establishes that the product of two sets with density 1 retains density 1 and provides counterexamples showing high-density sets can have low-density product sets, answering open questions.

Findings

Product sets of two density-1 sets have density 1.

Existence of high-density sets with low-density product sets.

Abstract

Given two sets of natural numbers $\mathcal{A}$ and $\mathcal{B}$ of natural density $1$ we prove that their product set $\mathcal{A}\cdot \mathcal{B}:=\{ab:a\in\mathcal{A},\,b\in\mathcal{B}\}$ also has natural density $1$. On the other hand, for any $\varepsilon>0$, we show there are sets $\mathcal{A}$ of density $>1-\varepsilon$ for which the product set $\mathcal{A}\cdot\mathcal{A}$ has density $<\varepsilon$. This answers two questions of Hegyv\'{a}ri, Hennecart and Pach.