On the extremal set of quotient of natural numbers
https://doi.org/10.22405/2226-8383-2023-24-4-354-360
Abstract
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 ∈ 𝐴, 𝑏 ∈ 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.
About the Author
Yuri Nikolaevich Shteinikov
Federal Research Center “Research Institute of System Research
of the Russian Academy of Sciences”
Russian Federation
Russian Federation
References
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Review
For citations:
Shteinikov Yu.N. On the extremal set of quotient of natural numbers. Chebyshevskii Sbornik. 2023;24(4):354-360. (In Russ.) https://doi.org/10.22405/2226-8383-2023-24-4-354-360
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