On the extremal set of quotient of natural numbers

Yu. N. Shteynikov

Čebyševskij sbornik, Tome 24 (2023) no. 4, pp. 354-360

Résumé

The article studies the following problem. Let two finite subsets from the set of natural numbers be given, which will be denoted throughout the text as $A$ and $B$. We will assume that they belong to a finite interval of numbers $[1,Q]$. By definition, we define a set of fractions $A/B$ whose elements are representable as a quotient of these sets $A,B$, in other words such elements $a/b$, where $a \in A, b \in B$. The article investigates the properties of this subset of quotients. In the article [13], a non-trivial lower bound on the size of the set $A/B$ for such sets $A,B$ was obtained without any additional conditions on these sets. In this article, we in details consider an extreme case, which is as follows. Let it be known that the size of the set of products $AB$ is asymptotically the smallest possible. We deduce from this that the size of the set of quotients $A/B$ is the asymptotically largest possible value.

Détail

Keywords: integer numbers, density, smooth numbers, product.