Long nn-zero-free sequences in finite cyclic groups

A sequence in the additive group ZnZn of integers modulo nn is called nn-zero-free if it does not contain subsequences with length nn and sum zero. The article characterizes the nn-zero-free sequences in ZnZn of length greater than 3n/2-13n/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 nn-zero-free sequence of any given length greater than 3n/2-13n/2-1 in ZnZn, 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.