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.
Categories: math.NT math.CO