On the additive bases problem in finite fields

TL;DR

This paper proves that in finite Abelian groups, a sum of sufficiently many subsets covering the group via multiple sums guarantees the entire group as a sumset, extending previous results on additive bases.

Contribution

It generalizes earlier work by establishing a logarithmic bound on the number of subsets needed for their sum to cover the entire group.

Findings

Sumsets of subsets cover the group when the number of subsets exceeds a logarithmic threshold.

Generalization of additive bases results to broader classes of Abelian groups.

Provides bounds that improve understanding of additive combinatorics in finite groups.

Abstract

We prove that if $G$ is an Abelian group and $A_1,\ldots,A_k \subseteq G$ satisfy $m A_i=G$ (the $m$-fold sumset), then $A_1+\ldots+A_k=G$ provided that $k \ge c_m \log n$. This generalizes a result of Alon, Linial, and Meshulam [Additive bases of vector spaces over prime fields. J. Combin. Theory Ser. A, 57(2):203--210, 1991] regarding the so called additive bases.