求助PDF
{"title":"正密度集合中的无穷集合","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":"{\"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}","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
引用
批量引用
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
