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| Natura: | Preprint |
| Pubblicazione: |
2025
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| Accesso online: | https://arxiv.org/abs/2510.20412 |
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| _version_ | 1866908607048581120 |
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| author | Girard, Benjamin Plagne, Alain |
| author_facet | Girard, Benjamin Plagne, Alain |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_20412 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Davenport constant of balls and boxes Girard, Benjamin Plagne, Alain Number Theory Combinatorics 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. |
| title | The Davenport constant of balls and boxes |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2510.20412 |