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Abstract <p> In the present paper, we improve the lower bound for the size of an integer set <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3048_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="17" /> </InlineMediaObject> <EquationSource Format="TEX">\(A\)</EquationSource> </InlineEquation> in a finite interval such that the cardinality of the product <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3048_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">\(AA\)</EquationSource> </InlineEquation> behaves asymptotically as <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11006_2025_3048_Article_IEq3.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\(|A|^2/2\)</EquationSource> </InlineEquation>. The main result presented here improves previous corresponding results of the papers [<CitationRef CitationID="CR1">1</CitationRef>] and [<CitationRef CitationID="CR2">2</CitationRef>] and uses additional arguments. </p>

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On Extremal Sets and Their Products in Finite Intervals of Integers

  • 期刊论文

  • 发表日期：

    2025年4月1日

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  • Yu. N. Shteinikov

摘要

Abstract

In the present paper, we improve the lower bound for the size of an integer set \(A\) in a finite interval such that the cardinality of the product \(AA\) behaves asymptotically as \(|A|^2/2\) . The main result presented here improves previous corresponding results of the papers [1] and [2] and uses additional arguments.