Zero subsums in vector spaces over finite fields

Abstract

The Olson constant $\mathcal{O}L({\mathbb{F}}_p^d)$ represents the minimum positive integer $t$ with the property that every subset $A\subset {\mathbb{F}}_p^d$ of cardinality $t$ contains a nonempty subset with vanishing sum. The problem of estimating $\mathcal{O}L({\mathbb{F}}_p^d)$ is one of the oldest questions in additive combinatorics, with a long and interesting history even for the case $d=1$. In this paper, we prove that for any fixed $d\ge 2$ and $\epsilon >0$, the Olson constant of ${\mathbb{F}}_p^d$ satisfies the inequality $$\mathcal{O}L({\mathbb{F}}_p^d)\le (d-1+\epsilon )p$$ for all sufficiently large primes $p$. This settles a conjecture of Hoi Nguyen and Van Vu.

References in corpus (3)

• Addition Theorems in Fp via the Polynomial Method
• An addition theorem and maximal zero-sum free sets in Z/pZ
• Convex geometry and the Erdős-Ginzburg-Ziv problem