关于列夫的周期性猜想

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2025-03-20 DOI:10.1112/blms.70043

Christian Reiher

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This yields a resolution of Vsevolod Lev's periodicity conjecture, which asserts that if a sum-free subset <math>\n <semantics>\n <mrow>\n <mi>A</mi>\n <mo>⊆</mo>\n <msubsup>\n <mi>F</mi>\n <mn>3</mn>\n <mi>n</mi>\n </msubsup>\n </mrow>\n <annotation>${A\\subseteq {\\mathbb {F}}_3^n}$</annotation>\n </semantics></math> is maximal with respect to inclusion and aperiodic (in the sense that there is no non-zero vector <math>\n <semantics>\n <mi>v</mi>\n <annotation>$v$</annotation>\n </semantics></math> satisfying <math>\n <semantics>\n <mrow>\n <mi>A</mi>\n <mo>+</mo>\n <mi>v</mi>\n <mo>=</mo>\n <mi>A</mi>\n </mrow>\n <annotation>$A+v=A$</annotation>\n </semantics></math>), then <math>\n <semantics>\n <mrow>\n <mrow>\n <mo>|</mo>\n <mi>A</mi>\n <mo>|</mo>\n </mrow>\n <mo>⩽</mo>\n <mfrac>\n <mn>1</mn>\n <mn>2</mn>\n </mfrac>\n <mrow>\n <mo>(</mo>\n <msup>\n <mn>3</mn>\n <mrow>\n <mi>n</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n </msup>\n <mo>+</mo>\n <mn>1</mn>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$|A|\\leqslant \\frac{1}{2}(3^{n-1}+1)$</annotation>\n </semantics></math>—a bound known to be optimal if <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>≠</mo>\n <mn>2</mn>\n </mrow>\n <annotation>$n\\ne 2$</annotation>\n </semantics></math>, while for <math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>=</mo>\n <mn>2</mn>\n </mrow>\n <annotation>$n=2$</annotation>\n </semantics></math> there are no such sets.","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 5","pages":"1496-1511"},"PeriodicalIF":0.8000,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70043","citationCount":"0","resultStr":"{\"title\":\"On Lev's periodicity conjecture\",\"authors\":\"Christian Reiher\",\"doi\":\"10.1112/blms.70043\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We classify the sum-free subsets of <math>\\n <semantics>\\n <msubsup>\\n <mi>F</mi>\\n <mn>3</mn>\\n <mi>n</mi>\\n </msubsup>\\n <annotation>${\\\\mathbb {F}}_3^n$</annotation>\\n </semantics></math> whose density exceeds <math>\\n <semantics>\\n <mfrac>\\n <mn>1</mn>\\n <mn>6</mn>\\n </mfrac>\\n <annotation>$\\\\frac{1}{6}$</annotation>\\n </semantics></math>. 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引用次数: 0

摘要

我们对密度超过16 $\frac{1}{6}$的f3n ${\mathbb {F}}_3^n$的无和子集进行了分类。这就解决了列夫的周期性猜想，它断言，如果一个无和的子集a≠F 3n ${A\subseteq {\mathbb {F}}_3^n}$在包含和非周期(即不存在非零向量v $v$满足？A + v = A $A+v=A$)，然后| A |≥1 2 （3 n−1）+ 1) $|A|\leqslant \frac{1}{2}(3^{n-1}+1)$ -当n≠2时已知的最优界$n\ne 2$，而对于n = 2 $n=2$，不存在这样的集合。

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On Lev's periodicity conjecture

We classify the sum-free subsets of $F_{3}^{n}$ whose density exceeds $\frac{1}{6}$ . This yields a resolution of Vsevolod Lev's periodicity conjecture, which asserts that if a sum-free subset $A \subseteq F_{3}^{n}$ is maximal with respect to inclusion and aperiodic (in the sense that there is no non-zero vector $v$ satisfying $A + v = A$ ), then $| A | ⩽ \frac{1}{2} (3^{n - 1} + 1)$ —a bound known to be optimal if $n \neq 2$ , while for $n = 2$ there are no such sets.