Sum-free sets which are closed under multiplicative inverses

Abstract

Let $A$ be a subset of a finite field $\mathbb{F}$. When $\mathbb{F}$ has prime order, we show that there is an absolute constant $c>0$ such that, if $A$ is both sum-free and equal to the set of its multiplicative inverses, then $|A|<(0.25–c)|\mathbb{F}|+o(|\mathbb{F}|)$ as $|\mathbb{F}|\to \mathrm{\infty }$. We contrast this with the result that such sets exist with size at least $0.25|\mathbb{F}|–o(|\mathbb{F}|)$ when $\mathbb{F}$ has characteristic 2.