  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.
