Sum-free sets which are closed under multiplicative inverses

TL;DR

This paper investigates the size limitations of subsets in finite fields that are both sum-free and closed under multiplicative inverses, establishing bounds in prime order fields and contrasting with characteristic 2 fields.

Contribution

It proves an upper bound on the size of such sets in prime order finite fields and highlights the existence of larger sets in characteristic 2 fields, revealing structural differences.

Findings

Sets in prime order fields are strictly less than 25% of the field size asymptotically.

Such sets can reach approximately 25% of the field size in characteristic 2 fields.

The results delineate how field characteristic influences the structure of sum-free inverse-closed sets.

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