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 𝒪L(𝔽pd) represents the minimum positive integer t with the property that every subset A 𝔽pd of cardinality t contains a nonempty subset with vanishing sum. The problem of estimating 𝒪L(𝔽pd) 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 2 and 𝜖 > 0, the Olson constant of 𝔽pd satisfies the inequality

𝒪L(𝔽pd) (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