On Lev's periodicity conjecture

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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Abstract

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.

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关于列夫的周期性猜想

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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