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arXiv · 2006.13356

A note on the natural density of product sets

Sandro Bettin·Dimitris Koukoulopoulos·Carlo Sanna
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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.

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Sandro Bettin, Dimitris Koukoulopoulos, Carlo Sanna. 2020-06-23. A note on the natural density of product sets. https://doi.org/10.1112/blms.12506

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