On products of sets of natural density one
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.
Authors
- Alessandro Fazzari (ORCID: https://orcid.org/0000-0002-6352-259X)
- Sandro Bettin (ORCID: https://orcid.org/0000-0003-3445-1603)
- Matteo Bordignon (ORCID: https://orcid.org/0000-0002-8347-0571)
Institutions
- Swedish e-Science Research Centre (SE)
- Istituto Nazionale di Fisica Nucleare, Sezione di Genova (IT)
- Université de Montréal (CA)
- KTH Royal Institute of Technology (SE)
Publication Details
- Journal
- Mathematical Proceedings of the Cambridge Philosophical Society
- Published
- 2026-04-07
- DOI
- https://doi.org/10.1017/s0305004126101893
- Primary Topic
- Limits and Structures in Graph Theory
- Type
- article
- Field-Weighted Citation Impact
- 0.00
Funders
- Istituto Nazionale di Alta Matematica "Francesco Severi"
- European Commission
- Ministero dell’Istruzione, dell’Università e della Ricerca
- Vetenskapsrådet
- Università degli Studi di Genova