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

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.

zero sum, Olson constant, finite fields, polynomial method
Mathematical Subject Classification
Primary: 05D40, 11P70
Received: 9 October 2020
Revised: 7 March 2021
Accepted: 17 August 2021
Published: 27 September 2022
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
Moscow Oblast