2025/10/09 by Manik Dhar, Dhar, Manik, Sammy Luo +1
Mathematics · #Analytic Number Theory Research #Mathematical Approximation and Integration #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2510.08857
Given a set A ⊆ \mathbbFpn, what conditions does one need to guarantee that iterated sumsets of the form A+⋯+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|>(1)/(p)|\mathbbFpn|. 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 A1+…+Am=\mathbbFpn as long as each Ai is not contained in the zero set of any low degree polynomial (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 Erdős-Ginzburg-Ziv problem. Our proofs make use of the shift operator polynomial method developed by the second author.