Long image-zero-free sequences in finite cyclic groups Original Research Article

A sequence in the additive group image of integers modulo image is called image-zero-free if it does not contain subsequences with length image and sum zero. The article characterizes the image-zero-free sequences in image of length greater than image. 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 image-zero-free sequence of any given length greater than image in image, 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.