Long n-zero-free sequences in finite cyclic groups

2006/04/16 by Svetoslav Savchev, Fang Chen, Savchev, Svetoslav +1
Mathematics · #11B50 #11P21 #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #Rings, Modules, and Algebras #math.CO #math.NT #msc:11B50 #msc:11P21

paper · pdf · doi:10.48550/arxiv.math/0604356

11 pages

arxiv created 2006/04/16 · openalex publication_date 2006/04/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A sequence in the additive group \mathbb Z_n of integers modulo n is called n-zero-free if it does not contain subsequences with length n and sum zero. The article characterizes the n-zero-free sequences in \mathbb Z_n of length greater than 3n/2-1. The structure of these sequences is completely determined, which generalizes a number of previously known facts. The characterization cannot be extended in the same form to shorter sequence lengths. Consequences of the main result are best possible lower bounds for the maximum multiplicity of a term in an n-zero-free sequence of any given length greater than 3n/2-1 in \mathbb Z_n, and also for the combined multiplicity of the two most repeated terms. Yet another application is finding the values in a certain range of a function related to the classic theorem of Erdős, Ginzburg and Ziv.

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