Given two sets of natural numbers (Formula presented.) and (Formula presented.) of natural density 1, we prove that their product set (Formula presented.) also has natural density 1. On the other hand, for any (Formula presented.), we show there are sets (Formula presented.) of density (Formula presented.) for which the product set (Formula presented.) has density (Formula presented.). This answers two questions of Hegyvári, Hennecart and Pach.
A note on the natural density of product sets
Bettin S.;
2021-01-01
Abstract
Given two sets of natural numbers (Formula presented.) and (Formula presented.) of natural density 1, we prove that their product set (Formula presented.) also has natural density 1. On the other hand, for any (Formula presented.), we show there are sets (Formula presented.) of density (Formula presented.) for which the product set (Formula presented.) has density (Formula presented.). This answers two questions of Hegyvári, Hennecart and Pach.
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