On zero-sum sequences of prescribed length

Summary.

Let k ≥  1 be any integer. Let G be a finite abelian group of exponent n. Let s _k (G) be the smallest positive integer t such that every sequence S in G of length at least t has a zero-sum subsequence of length kn. We study this constant for groups $G \cong {\user2{{\mathbb{Z}}}}^{d}_{n}$ when d = 3 or 4. In particular, we prove, as a main result, that $s_{k} ({\user2{{\mathbb{Z}}}}^{3}_{p} ) = kp + 3p - 3$ for every k ≥  4, $5p + \frac{{p - 1}}{2} - 3 \leq s_{2} ({\user2{{\mathbb{Z}}}}^{3}_{p} ) \leq 6p - 3$ and $6p - 3 \leq s_{3} ({\user2{{\mathbb{Z}}}}^{3}_{p} ) \leq 8p - 7$ for every prime p ≥  5.