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.
Bettin et al. (Tue,) studied this question.