Iterated sumset expansion in $\mathbb{F}_p^n$

Abstract

Given a set $A \subseteq \mathbb{F}_p^n$, what conditions does one need to guarantee that iterated sumsets of the form $A+\cdots+A$ expand quickly (say, within $O(p)$ terms) to the whole space? When only the size of $A$ is known, such expansion results are only possible when $|A|>\frac{1}{p}|\mathbb{F}_p^n|$. However, heuristic considerations suggest that expansion should begin with much smaller sets under just mild ``nondegeneracy'' conditions. In this paper, we confirm this intuition by showing a sufficient algebraic condition for the asymmetric version of this problem: We have $A_1+\dots+A_m=\mathbb{F}_p^n$ as long as each $A_i$ is not contained in the zero set of any low degree polynomial ($\text{deg} = O(n)$ when $m=O(p)$). We close with a discussion of the behavior of random sets, as well as extensions of these results and connections with the Erd\H{o}s-Ginzburg-Ziv problem. Our proofs make use of the shift operator polynomial method developed by the second author.