Bryna Kra, Joel Moreira, Florian Richter, Donald Robertson
{"title":"Infinite sumsets in sets with positive density","authors":"Bryna Kra, Joel Moreira, Florian Richter, Donald Robertson","doi":"10.1090/jams/1030","DOIUrl":null,"url":null,"abstract":"Motivated by questions asked by Erdős, we prove that any set <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A subset-of double-struck upper N\"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">A\\subset \\mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with positive upper density contains, for any <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k element-of double-struck upper N\"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">k\\in \\mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, a sumset <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B 1 plus midline-horizontal-ellipsis plus upper B Subscript k\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>+</mml:mo> <mml:mo>⋯<!-- ⋯ --></mml:mo> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">B_1+\\cdots +B_k</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B 1\"> <mml:semantics> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:annotation encoding=\"application/x-tex\">B_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, …, <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper B Subscript k Baseline subset-of double-struck upper N\"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi mathvariant=\"double-struck\">N</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">B_k\\subset \\mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are infinite. Our proof uses ergodic theory and relies on structural results for measure preserving systems. Our techniques are new, even for the previously known case of <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"k equals 2\"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">k=2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"22 1","pages":"0"},"PeriodicalIF":3.5000,"publicationDate":"2023-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the American Mathematical Society","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1090/jams/1030","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Motivated by questions asked by Erdős, we prove that any set A⊂NA\subset \mathbb {N} with positive upper density contains, for any k∈Nk\in \mathbb {N}, a sumset B1+⋯+BkB_1+\cdots +B_k, where B1B_1, …, Bk⊂NB_k\subset \mathbb {N} are infinite. Our proof uses ergodic theory and relies on structural results for measure preserving systems. Our techniques are new, even for the previously known case of k=2k=2.
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