A note on the natural density of 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 $\epsilon >0$, we show there are sets $\mathcal{A}$ of density $>1-\epsilon$ for which the product set $\mathcal{A}\cdot \mathcal{A}$ has density $<\epsilon$. This answers two questions of Hegyvári, Hennecart and Pach.

6 pages

References in corpus (1)

• A note on product sets of random sets