On the set of exceptions in the product of sets of natural numbers with asymptotic density 1

Abstract

The article examines the following problem. Let there be two subsets of the set of natural numbers, which we denote as 𝐴 and 𝐵. Let it also be additionally known that the asymptotic density of these sets 𝐴,𝐵 is 1. We define the set of natural numbers that are representable
as the product of these sets 𝐴𝐵, that is, such elements 𝑎𝑏, where 𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵. We study the properties of this subset of products in the set of all natural numbers. The authors S.
Bettin, D. Koukoulopoulos and C. Sanna in the article [1] proved, among other things, that the density of the set 𝐴𝐵 is also equal to 1. Moreover, a quantitative version of this statement was derived, namely, an estimate was obtained for the set N ∖ 𝐴𝐵, which we will denote by 𝐴𝐵. Namely, by these authors, in the case when quantitative upper bounds are known for 𝐴 ∩ [1, 𝑥] = 𝛼(𝑥)𝑥,𝐵 ∩ [1, 𝑥] = 𝛽(𝑥)𝑥, 𝛼(𝑥), 𝛽(𝑥) = 𝑂(1/(log 𝑥)𝑎), 𝑥 → ∞ the upper bound on the set 𝐴𝐵 ∩ [1, 𝑥] is also derived. In this paper, we study the case when 𝛼, 𝛽 tend to zero
slower than in the above case and somewhat refine the upper bound on the set 𝐴𝐵 ∩ [1, 𝑥]. In this paper we consider the case of 𝛼(𝑥), 𝛽(𝑥) = 𝑂(︀ 1/(log log 𝑥)𝑎)︀ for some fixed 𝑎 > 1. We borrow approaches, arguments and proof scheme from the mentioned work of three authors S. Bettin, D. Koukoulopoulos and C. Sanna[1].