Davenport constant of a box in $\mathbb {Z}^2$
Volume 197 / 2021
Acta Arithmetica 197 (2021), 259-274
MSC: Primary 11B75; Secondary 11B30, 11P70.
DOI: 10.4064/aa191010-15-8
Published online: 9 December 2020
Abstract
Let $X$ be a subset of an abelian group $G$. Then a sequence $S$ over $X$ is called a zero-sum sequence if the sum of $S$ is zero, and a minimal zero-sum sequence if it is a non-empty zero-sum sequence such that all proper subsequences are not zero-sum sequences. The Davenport constant of $X$, denoted by $\mathsf {D}(X)$, is defined as the supremum of lengths of minimal zero-sum sequences over $X$. In this paper, we investigate minimal zero-sum sequences over $[\![ -1,m]\!] \times [\![ -1,n]\!] \subset \mathbb {Z}^2$. We completely determine the structure of minimal zero-sum sequences of length at least $(m+1)(n+1)$, and hence derive that $\mathsf {D}([\![ -1,m]\!] \times [\![ -1,n]\!] )=(m+1)(n+1)$.
