Long $n$-zero-free sequences in finite cyclic groups

TL;DR

This paper characterizes long $n$-zero-free sequences in cyclic groups, providing structural insights and bounds, extending known results and connecting to classical theorems in additive combinatorics.

Contribution

It completely characterizes $n$-zero-free sequences of length greater than $3n/2-1$ in cyclic groups, generalizing previous results and deriving bounds on term multiplicities.

Findings

Structural description of $n$-zero-free sequences for length > 3n/2-1

Lower bounds for maximum term multiplicity in such sequences

Application to values related to Erdős-Ginzburg-Ziv theorem

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 Erd\H{o}s, Ginzburg and Ziv.