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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Sandro Bettin
Matteo Bordignon
Alessandro Fazzari
Mathematical Proceedings of the Cambridge Philosophical Society
Université de Montréal
KTH Royal Institute of Technology
Istituto Nazionale di Fisica Nucleare, Sezione di Genova
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Bettin et al. (Tue,) studied this question.
www.synapsesocial.com/papers/69d894ad6c1944d70ce059fa — DOI: https://doi.org/10.1017/s0305004126101893