Work

On products of sets of natural density one

Year: 2026

Type: article

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 upper A dot upper B colon equals left brace a b colon a element of upper A comma b element of upper B right brace A ⋅ B : = { a b : a ∈ A , b ∈ B } $A \cdot B \;:\!=\; \{ab \;:\; a \in A, b \in B\}$ 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 upper A dot upper A A ⋅ A $A\cdot A$ has a “large” complement.

Source: Mathematical Proceedings of the Cambridge Philosophical Society

DOI: 10.1017/s0305004126101893

Authors: SANDRO BETTIN, MATTEO BORDIGNON, ALESSANDRO FAZZARI

Institutions: University of Genoa, KTH Royal Institute of Technology, Université de Montréal

Language: English

Topic: Analytic Number Theory Research

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,550