Zero-sum subsets 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 $d=1$.

We prove that for any fixed $d\ge 2$ and , 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.