The Davenport constant of balls and boxes

TL;DR

The paper initiates a systematic study of the Davenport constant for subsets of abelian groups, focusing on discrete Euclidean balls $\mathcal{B}^{(d)}_m$ in $\mathbb{Z}^d$ and the related box problem. It develops a framework using minimal zero-sum sequences, refined invariants $\mathsf{D}^{(k)}(X)$, and Steinitz-type convex-geometry arguments to obtain tight bounds and asymptotics in low dimensions, notably $d=2$ and $d=3$, and to connect ball and box constants. In dimension two, it determines a precise lower bound and an almost tight upper bound for $\mathsf{D}(\mathcal{B}^{(2)}_m)$, with exact asymptotics for $\mathsf{D}^{(3)}(\mathcal{B}^{(2)}_m)$ and a sharp formula for $\mathsf{D}^{(3)}(\llbracket -m,m \rrbracket^2)$ involving the function $q(m)$, namely $\mathsf{D}^{(3)}(\llbracket -m,m \rrbracket^2)=4m^2-q(m)$. In dimension three, the authors establish substantial lower and upper bounds for $\mathsf{D}(\mathcal{B}^{(3)}_m)$ and prove optimality for $\mathsf{D}^{(4)}(\mathcal{B}^{(3)}_m)$, showing $\mathsf{D}^{(4)}(\mathcal{B}^{(3)}_m) \sim \frac{16}{3\sqrt{3}} m^3$ by identifying the rhombic dodecahedron as the maximizing polyhedron, whose visibility arises from a regular tetrahedron in the proof. They also propose a broad conjecture for general $d$, predicting $\mathsf{D}(\mathcal{B}^{(d)}_m)$ and $\mathsf{D}^{(d+1)}(\mathcal{B}^{(d)}_m)$ scale like $\sqrt{(d+1)^{d+1}/d^d}\,m^d$ with extremal configurations given by $d+1$ points in regular simplex position on the unit sphere. The work connects additive combinatorics with convex geometry, yielding new bounds for balls and boxes and advancing the understanding of extremal zero-sum phenomena in lattice settings.

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.

Figures (4)

• Figure 1: A representation of $H(w_1,w_2,w_3)$.
• Figure 2: Two decompositions of the hexagon $H(w_1, w_2, w_3)$.
• Figure 3: A representation of $D(w_1,w_2,w_3,w_4)$ as a translucent polyhedron.
• Figure 4: Each of the two figures represents $D(w_1,w_2,w_3,w_4)$ as an opaque polyhedron. On the left, the three visible faces of $D_2$ are in white, the only visible face of $D_3$ in light grey and the two visible faces of $D_4$ in medium grey ($D_1$ is entirely hidden at the back of the solid). On the right, the three visible faces of $w_1+D_1$ are in dark grey, the only visible face of $w_4+D_4$ in medium grey and the two visible faces of $w_3+D_3$ in light grey ($w_2+D_2$ is entirely hidden at the back of the solid).

Theorems & Definitions (33)

• Theorem 1
• Theorem 2
• Theorem 3
• Conjecture 1
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Theorem 8
• Conjecture 2