The Davenport constant of balls and boxes

Abstract

Given an additively written abelian group $G$ and a set $X\subseteq G$, we let $\mathsf{D}(X)$ denote the Davenport constant of $X$, namely the largest non-negative integer $n$ for which there exists a sequence $x_1,\dots ,x_n$ of elements of $X$ such that ${\sum }_{i=1}^n{x}_i=0$ and ${\sum }_{i\in I}{x}_i\ne 0$ for each non-empty proper subset $I$ of $\{1,\dots ,n\}$. In this paper, we mainly investigate the case when $G$ is ${\mathbb{Z}}^2$ and ${\mathbb{Z}}^3$, and $X$ is a discrete Euclidean ball. An application to the classical problem of estimating the Davenport constant of a box - a product of intervals of integers - is then obtained.

43 pages, 4 figures