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arXiv · 2504.01665

On products of sets of natural density one

Sandro Bettin·Matteo Bordignon·Alessandro Fazzari
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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 $A \cdot B := \{ab : a \in A, b \in B\}$ 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 $A\cdot A$ has a ''large'' complement.

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Sandro Bettin, Matteo Bordignon, Alessandro Fazzari. 2025-04-02. On products of sets of natural density one. https://arxiv.org/abs/2504.01665

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