The $\{1,s\}$-weighted Davenport constant in $C_n^k$ and an application in an inverse problem
Year: 2018
Type: preprint
Abstract: Let $G$ be a finite abelian group and $\varnothing \neq A \subset \mathbb Z$. The $A$-weighted Davenport constant of $G$, ${\sf D}_A(G)$, is the smallest positive integer $D$ such that every sequence $x_1 \bullet \dots \bullet x_D$ over $G$ has a non-empty subsequence $(x_{j_i})_i$ such that $$x_{j_1}^{\varepsilon_1} \cdot x_{j_2}^{\varepsilon_2} \cdot \dots \cdot x_{j_t}^{\varepsilon_t} = 1$$ for some $\varepsilon_1, \varepsilon_2, \dots, \varepsilon_t \in A$. In this paper, we obtain both upper and lower bounds for ${\sf D}_{\{1,s\}}(C_n^k)$, where $C_n$ denotes the cyclic group of order $n$ and $s^2 - 1$ is divisible by $2n$ if $n$ is even and divisible by $n$ if $n$ is odd, but $s \not\equiv \pm1 \pmod n$. These bounds become sharp in some small cases. As application, we determine all extremal length product-one free sequences over $C_n \rtimes_s C_2$.
Source: arXiv (Cornell University)
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2795342113