This research demonstrates the properties of density one sets, revealing a large complement in their product set.
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 \;:\!=\; \;:\; 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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Bettin et al. (2026) studied this question.
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