For set $A\subset {\mathbb {F}_p}^*$ define by ${\mathsf{sf}}(A)$ the size of the largest sum--free subset of $A.$ Alon and Kleitman showed that ${\mathsf{sf}} (A) \ge |A|/3+O(|A|/p).$ We prove that if ${\mathsf{sf}} (A)-|A|/3$ is small then the set $A$ must be uniformly distributed on cosets of each large multiplicative subgroup. Our argument relies on irregularity of distribution of multiplicative subgroups on certain intervals in ${\mathbb {F}_p}$.

中文翻译：

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L 函数和无和集

对于集合 $A\subset {\mathbb {F}_p}^*$ 由 ${\mathsf{sf}}(A)$ 定义 $A 的最大无和子集的大小。 ${\mathsf{sf}} (A) \ge |A|/3+O(|A|/p).$ 我们证明如果 ${\mathsf{sf}} (A)-|A|/ 3$ 很小，则集合 $A$ 必须均匀分布在每个大乘法子群的陪集上。我们的论点依赖于 ${\mathbb {F}_p}$ 中某些区间上乘法子群分布的不规则性。