#### Vol. 16, No. 6, 2022

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Zero-sum subsets in vector spaces over finite fields

### Cosmin Pohoata and Dmitriy Zakharov

Vol. 16 (2022), No. 6, 1407–1421
##### Abstract

The Olson constant $\mathsc{𝒪}L\left({\mathbb{𝔽}}_{p}^{d}\right)$ 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 $\mathsc{𝒪}L\left({\mathbb{𝔽}}_{p}^{d}\right)$ 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 $𝜖>0$, the Olson constant of ${\mathbb{𝔽}}_{p}^{d}$ satisfies the inequality

 $\mathsc{𝒪}L\left({\mathbb{𝔽}}_{p}^{d}\right)\le \left(d-1+𝜖\right)p$

for all sufficiently large primes $p$. This settles a conjecture of Hoi Nguyen and Van Vu.

##### Keywords
zero sum, Olson constant, finite fields, polynomial method
##### Mathematical Subject Classification
Primary: 05D40, 11P70