Long minimal zero-sum sequences over a finite subset of Z

Let G be an abelian group (written additively), X be a subset of G and S be a minimal zero-sum sequence over X. S is called unsplittable in X if there do not exist an element g in S and two elements x,y in X such that g=x+y and the new sequence Sg−1xy is still a minimal zero-sum sequence. In this paper, we mainly investigate the case when G=Z and X=〚−m,n〛 with m,n∈N. We obtain the structure of unsplittable minimal zero-sum sequences of length at least n+⌊m∕2⌋+2 provided that n≥m2∕2−1 and m≥6. As a corollary, the Davenport constant D(〚−m,n〛) is determined when n≥m2∕2−1. The Davenport constant D(X) for a general set X⊂Z is also discussed.