On products of sets of natural density one

2025/04/02 by Bettin, Sandro, Bordignon, Matteo, Fazzari, Alessandro
#11B05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2504.01665

Abstract

In a previous work, Bettin, Koukoulopoulos, and Sanna prove that if two sets of natural numbers A and B have natural density 1, then their product set A ⋅ B := \ab : a ∈ A, b ∈ B\ also has natural density 1. They also provide an effective rate and pose the question of determining the optimal rate. We make progress on this question by constructing a set A of density 1 such that A⋅ A has a ''large'' complement.

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