On special extremal sets associated with the multiplication table of P. Erdős

Yu. N. Shteinikov

Čebyševskij sbornik, Tome 25 (2024) no. 3, pp. 373-380

Résumé

This article investigates the following problem arising from the theory of products of sets. Let there be two finite subsets of the set of natural numbers, which throughout the article will be denoted as $A$ and $B$. We assume that they are a subset of the interval of numbers $[1,Q]$. By definition, we introduce a set called the product set $AB$, the elements of which are represented as a product of elements from $A,B$, in other words, such elements $ab$, where $a \in A, b \in B$. This article studies the problem of extremely large sets $A$ of a finite interval $[1,Q]$ that have the asymptotically largest possible product, that is, the asymptotically largest value of $|AA|$ equal to $|A|^2/2$. In the paper [2], a new non-trivial lower bound for the size of such a set $A$ was obtained in comparison with the previous result of the paper by K. Ford [Shteinikov_base] and also of the paper [1]. In this article we present a method that improves the previous result, and also introduce another version of this problem. In general, we follow and develop the formulations, arguments, ideas and approaches proposed in the works [1], [2].

Détail

Keywords: integer numbers, density, product.