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April 10, 2026Mathematical Proceedings of the Cambridge Philosophical Society0 citationsOpen Access

On products of sets of natural density one

SBSandro BettinMBMatteo BordignonAFAlessandro Fazzari

Key Points

  • The aim is to explore the relationship between the natural density of sets and the density of their product sets.
  • Constructed a set A of natural numbers with density 1.
  • Analyzed the product set A·A and its complement.
  • Investigated the effective rate related to the product's density.
  • Demonstrated that the constructed set A·A has a ‘large’ complement.
  • Confirmed that A·A maintains a natural density of 1.
  • Provided insight into the optimal rate concerning density.

Abstract

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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Cite This Study

Bettin et al. (2026) studied this question.

synapsesocial.com/papers/69d894ad6c1944d70ce059fahttps://doi.org/10.1017/s0305004126101893
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