The Davenport constant of a box

Alain Plagne ¹   ; Salvatore Tringali ²

¹ Centre de math\'ematiques Laurent Schwartz \'Ecole polytechnique 91128 Palaiseau Cedex, France
² Science Program Texas A\&M University at Qatar Education City PO Box 23874, Doha, Qatar

Acta Arithmetica, Tome 171 (2015) no. 3, pp. 197-219

Résumé

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 $x_1 \cdots x_n$ in $\mathscr{B}(X)$ 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.

Détail

DOI : 10.4064/aa171-3-1

Keywords: given additively written abelian group set subseteq mathscr denote monoid zero sum sequences mathsf davenport constant mathscr namely supremum positive integers which there exists sequence cdots mathscr sum each non empty proper subset ldots paper mainly investigate power mathbb box product intervals mixed sets product group box studied too inverse results obtained