关于移位素数的萨科齐定理

IF 3.5 1区数学 Q1 MATHEMATICS Journal of the American Mathematical Society Pub Date : 2023-09-28 DOI:10.1090/jams/1036

Ben Green

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Then <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartAbsoluteValue upper A EndAbsoluteValue much-less-than upper N Superscript 1 minus c\"> <mml:semantics> <mml:mrow> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mi>A</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mo stretchy=\"false\">|</mml:mo> </mml:mrow> <mml:mo>≪</mml:mo> <mml:msup> <mml:mi>N</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>c</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">|A| \\ll N^{1 - c}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"48 1","pages":"0"},"PeriodicalIF":3.5000,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Sarkozy’s theorem for shifted primes\",\"authors\":\"Ben Green\",\"doi\":\"10.1090/jams/1036\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Suppose that <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper A subset-of StartSet 1 comma ellipsis comma upper N EndSet\\\"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊂</mml:mo> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">{</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mi>N</mml:mi> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">A \\\\subset \\\\{1,\\\\dots , N\\\\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has no two elements differing by <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"p minus 1\\\"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">p-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=\\\"p\\\"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> prime. Then <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"StartAbsoluteValue upper A EndAbsoluteValue much-less-than upper N Superscript 1 minus c\\\"> <mml:semantics> <mml:mrow> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mo stretchy=\\\"false\\\">|</mml:mo> </mml:mrow> <mml:mi>A</mml:mi> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mo stretchy=\\\"false\\\">|</mml:mo> </mml:mrow> <mml:mo>≪</mml:mo> <mml:msup> <mml:mi>N</mml:mi> <mml:mrow class=\\\"MJX-TeXAtom-ORD\\\"> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>c</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">|A| \\\\ll N^{1 - c}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.\",\"PeriodicalId\":54764,\"journal\":{\"name\":\"Journal of the American Mathematical Society\",\"volume\":\"48 1\",\"pages\":\"0\"},\"PeriodicalIF\":3.5000,\"publicationDate\":\"2023-09-28\",\"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/1036\",\"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/1036","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

摘要

想那A ...⊂{1,N}子集\{1,圆点,N \号的有两个文本differing by−1 p - 1, p p p prime。然后| A |≪N−1 | c . | \会1 - c N ^{}。

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On Sarkozy’s theorem for shifted primes

Suppose that A ⊂ { 1 , … , N } A \subset \{1,\dots , N\} has no two elements differing by p − 1 p-1 , p p prime. Then | A | ≪ N 1 − c |A| \ll N^{1 - c} .

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来源期刊

Journal of the American Mathematical Society 数学-数学

CiteScore

7.60

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： All articles submitted to this journal are peer-reviewed. The AMS has a single blind peer-review process in which the reviewers know who the authors of the manuscript are, but the authors do not have access to the information on who the peer reviewers are. This journal is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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