The Davenport constant of a box

TL;DR

This paper investigates the Davenport constant for zero-sum sequences over boxes in abelian groups, especially powers of integers, providing new bounds, inverse results, and studying mixed sets involving products of groups and boxes.

Contribution

It introduces new results on the Davenport constant for boxes in abelian groups, including bounds and inverse theorems, expanding understanding of zero-sum sequence properties in these structures.

Findings

Derived bounds for the Davenport constant of boxes in abelian groups

Studied inverse problems related to zero-sum sequences over boxes

Analyzed mixed sets involving products of groups and boxes

Abstract

Given an additively written abelian group $G$ and a set $X\subseteq G$, we let $\mathscr{B}(X)$ denote the monoid of zero-sum sequences over $X$ and $\mathsf{D}(X)$ the Davenport constant of $\mathscr{B}(X)$, namely the supremum of the positive integers $n$ for which there exists a sequence $x_1 \cdots x_n$ of $\mathscr{B}(X)$ such that $\sum_{i \in I} x_i \ne 0$ for each non-empty proper subset $I$ of $\{1, \ldots, n\}$. In this paper, we mainly investigate the case when $G$ is a power of $\mathbb{Z}$ and $X$ is a box (i.e., a product of intervals of $G$). Some mixed sets (e.g., the product of a group by a box) are studied too, and some inverse results are obtained.