$L$--functions and sum--free sets

Abstract

For set $A\subset {{\mathbb{F}}_p}^{\ast }$ 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 ${\mathbb{F}}_p$.

15 pages