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

For set $A\subset {{\mathbb{F}}_p}^{\ast }$ define by $\mathsf{s}\mathsf{f}(A)$ the size of the largest sum–free subset of $A\mathrm{.}$ Alon and Kleitman showed that $\mathsf{s}\mathsf{f}(A)\ge \mathrm{\mid }A\mathrm{\mid }\mathrm{/}3+O(\mathrm{\mid }A\mathrm{\mid }\mathrm{/}p)\mathrm{.}$ We prove that if ${$\mathsf{s}\mathsf{f}$} (A)-|A|/3$issmallthentheset$A$ 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 ${\mathbb {F}_p}$.