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

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 $a\in A, b\in B$. 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 $\mathbb{N}\setminus AB$, which we will denote by $\overline{AB}$. Namely, by these authors, in the case when quantitative upper bounds are known for $\overline{A}\cap[1,x] =\alpha(x)x, \overline{B}\cap[1,x] = \beta(x)x, \alpha(x),\beta(x) = O(1/(\log x)^a), x\rightarrow \infty$ the upper bound on the set $\overline{AB}\cap [1,x]$ is also derived. In this paper, we study the case when $\alpha, \beta$ tend to zero slower than in the above case and somewhat refine the upper bound on the set $\overline{AB}\cap[1,x]$. In this paper we consider the case of $\alpha(x), \beta(x) = O\bigl(\frac{1}{(\log\log x)^a}\bigr)$ 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].