The Davenport constant of balls and boxes

TL;DR

This paper investigates the Davenport constant for discrete Euclidean balls within groups like  and , and applies findings to estimate the Davenport constant of integer boxes, advancing understanding in additive combinatorics.

Contribution

It provides new bounds and insights into the Davenport constant for Euclidean balls in  and , and connects these results to the classical problem of boxes in integer groups.

Findings

Determined the Davenport constant for Euclidean balls in  and .

Established bounds for the Davenport constant of boxes of integers.

Linked geometric properties of Euclidean balls to additive combinatorics results.

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 $\{1, \ldots, n\}$. 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.