On Short Zero-Sum Subsequences Over $p$-Groups

Abstract

Let $G$ be a finite abelian group with exponent $n$. Let $s(G)$ denote the smallest integer $l$ such that every sequence over $G$ of length at least $l$ has a zero-sum subsequence of length $n$. For $p$-groups whose exponent is odd and sufficiently large (relative to Davenport’s constant of the group) we obtain an improved upper bound on $s(G)$, which allows to determine $s(G)$ precisely in special cases. Our results contain Kemnitz’ conjecture, which was recently proved, as a special case.