A characterization of incomplete sequences in $F_p^d$

Abstract

A sequence $A$ of elements an additive group $G$ is {\it incomplete} if there exists a group element that {\it can not} be expressed as a sum of elements from $A$. The study of incomplete sequences is a popular topic in combinatorial number theory. However, the structure of incomplete sequences is still far from being understood, even in basic groups. The main goal of this paper is to give a characterization of incomplete sequences in the vector space $F_p^d$, where $d$ is a fixed integer and $p$ is a large prime. As an application, we give a new proof for a recent result by Gao-Ruzsa-Thangadurai on the Olson's constant of $\F_p^2$ and partially answer their conjecture concerning $F_p^3$.