Sequences not containing long zero-sum subsequences

Let GG be a finite abelian group (written additively), and let D(G)D(G) denote the Davenport’s constant of GG, i.e. the smallest integer dd such that every sequence of dd elements (repetition allowed) in GG contains a nonempty zero-sum subsequence. Let SS be a sequence of elements in GG with |S|≥D(G)|S|≥D(G). We say SS is a normal sequence if SS contains no zero-sum subsequence of length larger than |S|−D(G)+1|S|−D(G)+1. In this paper we obtain some results on the structure of normal sequences for arbitrary GG. If G=Cn⊕CnG=Cn⊕Cn and nn satisfies some well-investigated property, we determine all normal sequences. Applying these results, we obtain correspondingly some results on the structure of the sequence SS in GG of length |S|=|G|+D(G)−2|S|=|G|+D(G)−2 and SS contains no zero-sum subsequence of length |G||G|.