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2022 Zero-sum subsets in vector spaces over finite fields
Cosmin Pohoata, Dmitriy Zakharov
Algebra Number Theory 16(6): 1407-1421 (2022). DOI: 10.2140/ant.2022.16.1407

Abstract

The Olson constant 𝒪L(𝔽pd) represents the minimum positive integer t with the property that every subset A𝔽pd 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 d2 and ε>0, the Olson constant of 𝔽pd satisfies the inequality

𝒪L(𝔽pd)(d1+ε)p

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

Citation

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Cosmin Pohoata. Dmitriy Zakharov. "Zero-sum subsets in vector spaces over finite fields." Algebra Number Theory 16 (6) 1407 - 1421, 2022. https://doi.org/10.2140/ant.2022.16.1407

Information

Received: 9 October 2020; Revised: 7 March 2021; Accepted: 17 August 2021; Published: 2022
First available in Project Euclid: 2 October 2022

Digital Object Identifier: 10.2140/ant.2022.16.1407

Subjects:
Primary: 05D40 , 11P70

Keywords: finite fields , Olson constant , polynomial method , zero sum

Rights: Copyright © 2022 Mathematical Sciences Publishers

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Vol.16 • No. 6 • 2022
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