Zero-sum subsets in vector spaces over finite fields

Pohoata, Cosmin; Zakharov, Dmitriy

doi:10.2140/ant.2022.16.1407

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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

DOI: 10.2140/ant.2022.16.1407

Abstract

The Olson constant $𝒪 L (𝔽_{p}^{d})$ represents the minimum positive integer $t$ with the property that every subset $A \subset 𝔽_{p}^{d}$ of cardinality $t$ contains a nonempty subset with vanishing sum. The problem of estimating $𝒪 L (𝔽_{p}^{d})$ 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 $d \geq 2$ and $𝜖 > 0$ , the Olson constant of $𝔽_{p}^{d}$ satisfies the inequality

𝒪 L (𝔽_{p}^{d}) \leq (d - 1 + 𝜖) 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

Milestones

Received: 9 October 2020

Revised: 7 March 2021

Accepted: 17 August 2021

Published: 27 September 2022

Authors

Cosmin Pohoata
Department of Mathematics Yale University New Haven, CT United States

Dmitriy Zakharov
Laboratory of Combinatorial and Geometric Structures Moscow Institute of Physics and Technology Dolgoprudny Moscow Oblast Russia

Vol. 16, No. 6, 2022