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 $|\mathbb{F}|\to \infty$. 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 $2$.

12 pages, 1 figure. Some computations are omitted but can be accessed through the source file