On the Davenport constant of a two-dimensional box $\left[\kern-0.15em\left[ { - 1, 1} \right]\kern-0.15em\right] \times \left[\kern-0.15em\left[ { - m, n} \right]\kern-0.15em\right]$

Guixin Deng; Shuxin Wang; Guixin Deng; Shuxin Wang

doi:10.3934/math.2021066

AIMS Mathematics

2021, Volume 6, Issue 2: 1101-1109. doi: 10.3934/math.2021066

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Research article

On the Davenport constant of a two-dimensional box $\left[\kern-0.15em\left[ { - 1, 1} \right]\kern-0.15em\right] \times \left[\kern-0.15em\left[ { - m, n} \right]\kern-0.15em\right]$

Guixin Deng ^,,
Shuxin Wang

College of Mathematics and Statistics, Nanning Normal University, Nanning, 530000, China

Received: 10 September 2020 Accepted: 28 October 2020 Published: 10 November 2020
MSC : 11B30, 11P70

Let $G$ be an abelian group and $X$ be a nonempty subset of $G$. A sequence $S$ over $X$ is called zero-sum if the sum of all terms of $S$ is zero. A nonempty zero-sum sequence $S$ is called minimal zero-sum if all nonempty proper subsequences of $S$ are not zero-sum. The Davenport constant of $X$, denoted by $\textsf{D}(X)$, is defined to be the supremum of lengths of all minimal zero-sum sequences over $X$. In this paper, we study the minimal zero-sum sequences over $X = \left[\kern-0.15em\left[ { - 1, 1} \right]\kern-0.15em\right] \times \left[\kern-0.15em\left[ { - m, n} \right]\kern-0.15em\right]\subset\mathbb{Z}^2$. We completely determine the structure of minimal zero-sum sequences of maximal length over $X$ and obtain that $\textsf{D}(X) = 2(n+m).$
- zero-sum,
- minimal zero-sum,
- Davenport constant
Citation: Guixin Deng, Shuxin Wang. On the Davenport constant of a two-dimensional box $\left[\kern-0.15em\left[ { - 1, 1} \right]\kern-0.15em\right] \times \left[\kern-0.15em\left[ { - m, n} \right]\kern-0.15em\right]$[J]. AIMS Mathematics, 2021, 6(2): 1101-1109. doi: 10.3934/math.2021066

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Abstract

Let $G$ be an abelian group and $X$ be a nonempty subset of $G$. A sequence $S$ over $X$ is called zero-sum if the sum of all terms of $S$ is zero. A nonempty zero-sum sequence $S$ is called minimal zero-sum if all nonempty proper subsequences of $S$ are not zero-sum. The Davenport constant of $X$, denoted by $\textsf{D}(X)$, is defined to be the supremum of lengths of all minimal zero-sum sequences over $X$. In this paper, we study the minimal zero-sum sequences over $X = \left[\kern-0.15em\left[ { - 1, 1} \right]\kern-0.15em\right] \times \left[\kern-0.15em\left[ { - m, n} \right]\kern-0.15em\right]\subset\mathbb{Z}^2$. We completely determine the structure of minimal zero-sum sequences of maximal length over $X$ and obtain that $\textsf{D}(X) = 2(n+m).$

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