On special extremal sets associated with the multiplication table of P. Erd˝os
https://doi.org/10.22405/2226-8383-2024-25-3-373-380
Abstract
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 ∈ 𝐴, 𝑏 ∈ 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 [1] and also of the paper [2]. 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].
About the Author
Yuri Nikolayevich Shteinikov
National Research Center «Kurchatov Institute»; Federal Research Center “Research Institute of System Research of the RAS”
Russian Federation
Russian Federation
References
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Review
For citations:
Shteinikov Yu.N. On special extremal sets associated with the multiplication table of P. Erd˝os. Chebyshevskii Sbornik. 2024;25(3):373-380. (In Russ.) https://doi.org/10.22405/2226-8383-2024-25-3-373-380
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