Structure theory of set addition with two operations

TL;DR

This paper develops a structure theory for sets in finite fields that exhibit specific additive behaviors under multiple power-based operations, extending classical additive combinatorics to a ring-like context.

Contribution

It introduces inverse results characterizing sets with constrained sumset sizes involving multiple power operations, advancing the understanding of ring structures in finite fields.

Findings

Identifies conditions under which sumsets involving powers are small

Provides inverse theorems linking set structure to sumset size

Extends additive combinatorics to include ring operations

Abstract

We take the first step toward a structure theory that includes both operations of a ring $\mathcal{R}$. More precisely, we prove a series of inverse results for the structure of sets $A\subseteq \mathbf{F}_p$ such that, under certain conditions on integers $r_1, \dots, r_k$, one has $|A^{r_1} + \dots + A^{r_k}| \ll \sqrt[k]{p^{k-1} |A|}$.