A note on the natural density of product sets

Abstract

Given two sets of natural numbers A and B of natural density 1, we prove that their product set A • B := {ab : a ∈ A, b ∈ B} also has natural density 1. On the other hand, for any ε > 0, we show there are sets A of density > 1 − ε for which the product set A • A has density < ε. This answers two questions of Hegyvári, Hennecart and Pach.