A characterization of incomplete sequences in $F_p^d$

TL;DR

This paper characterizes incomplete sequences in the vector space over a finite field, providing new insights into their structure and applications to Olson's constant in low-dimensional cases.

Contribution

It offers a novel characterization of incomplete sequences in $F_p^d$, advancing understanding in combinatorial number theory and addressing open conjectures.

Findings

New characterization of incomplete sequences in $F_p^d$

Simplified proof of Olson's constant for $_p^2$

Partial progress on Olson's conjecture for $_p^3$

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 $F_p^3$.