A note on the natural density of product sets

Given two sets of natural numbers $\mathcal{A}$ and $\mathcal{B}$ of natural density $1$ we prove that their product set $$\mathcal{A}$\cdot $\mathcal{B}$:={ab:a\in$\mathcal{A}$,,b\in$\mathcal{B}$}$alsohasnaturaldensity$1$\mathrm{.}Ontheotherhand,forany$\varepsilon>0$,weshowtherearesets$$\mathcal{A}$$ofdensity$>1-\varepsilon$forwhichtheproductset$$\mathcal{A}$\cdot$\mathcal{A}$$hasdensity$<\varepsilon$. This answers two questions of Hegyv'{a}ri, Hennecart and Pach.