Zero subsums in vector spaces over finite fields

TL;DR

This paper establishes an upper bound for the Olson constant in vector spaces over finite fields, confirming a conjecture and advancing understanding in additive combinatorics.

Contribution

It proves that for fixed dimension, the Olson constant grows linearly with the prime size, settling a longstanding conjecture.

Findings

Olson constant is bounded above by (d-1+ε)p for large primes p.

Confirms a conjecture of Nguyen and Vu.

Advances understanding of zero-sum problems in finite vector spaces.

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 $d \geq 2$ and $\epsilon > 0$, the Olson constant of $\mathbb{F}_{p}^{d}$ satisfies the inequality $$\mathcal{O}L(\mathbb{F}_{p}^{d}) \leq (d-1+\epsilon)p$$ for all sufficiently large primes $p$. This settles a conjecture of Hoi Nguyen and Van Vu.