The Davenport constant of low-dimensional boxes

Abstract

In this paper, we study the maximal length of a minimal zero-sum sequence over $\left[\!\left[ -m,m \right]\!\right]^d$ having a support of size $d+1$. When $d=2$, we find the exact value of this maximal length for all $m \ge 2$, as well as the exact structure of all extremal sequences for this problem. When $d=3$, we obtain lower and upper bounds that asymptotically match each other as $m$ tends to infinity. This allows us to improve on the best known lower bounds for the Davenport constant of low-dimensional boxes.