Zero Sums in Finite Cyclic Groups

Let Cn be the cyclic group of n elements, and let S = (a1, · · · , ak) be a sequence of elements in Cn. We say that S is a zero sequence if ∑k i=1 ai = 0 and that S is a minimal zero-sequence if S is a zero sequence and S contains no proper zero subsequence. In this paper we prove, among other results, that if S is a minimal zero sequence of elements in Cn and |S| ≥ n − [ 3 ] + 1, then there exists an integer m coprime to n such that |ma1|+ · · ·+ |mak| = n, where |x| denotes the least positive inverse image under the natural homomorphism from the additive group of integers Z onto Cn. On the other hand, we give some explicit minimal zero sequences of length [ 2 ] + 1 not having this property above.