On the Additive Bases Problem in Finite Fields

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 $A_1+\cdots+A_k=G$ provided that $k \ge c_m \log \log |G|$. 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 so-called additive bases.