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<p>In the study of sums of finite sets of integers, most attention has been paid to sets with small sumsets (Freiman's theorem and related work) and to sets with large sumsets (Sidon sets and <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1559_Article_IEq10.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\(B_h\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>h</mi> </msub> </math></EquationSource> </InlineEquation>-sets). This paper focuses on the full range of sizes of <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1559_Article_IEq11.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">\(h\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>h</mi> </math></EquationSource> </InlineEquation>-fold sums of a set of <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2025_1559_Article_IEq12.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(k\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>k</mi> </math></EquationSource> </InlineEquation> integers. Many new results and open problems are presented.</p>

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Problems in additive number theory. VI: Sizes of sumsets of finite sets

  • 期刊论文

  • 发表日期：

    2025年8月1日

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  • M. B. Nathanson

摘要

In the study of sums of finite sets of integers, most attention has been paid to sets with small sumsets (Freiman's theorem and related work) and to sets with large sumsets (Sidon sets and \(B_h\) B h -sets). This paper focuses on the full range of sizes of \(h\) h -fold sums of a set of \(k\) k integers. Many new results and open problems are presented.