Some Davenport Constants with Weights and Adhikari $\mathrm{\&}$ Rath’s Conjecture

Abstract

Let $n\in \mathbb{N}$ and let $A\subseteq {\mathbb{Z}}_n$ be such that $A$ does not contain $0$ and is non-empty. We define $E_A(n)$ to be the least $t\in \mathbb{N}$ such that for all sequences $(x_1,\dots ,x_t)\in {\mathbb{Z}}^t$, there exist indices $j_1,\dots ,j_n\in \mathbb{N}$, $1\le j_1<\cdots <j_n\le t$, and $({\theta }_1,\dots ,{\theta }_n)\in A^n$ with $\sum _{i=1}^n{\theta }_i{x}_{j_i}\equiv 0(\mathrm{mod}n)$. Similarly, for any such set $A$, we define the $DavenportConstant$ of ${\mathbb{Z}}_n$ with weight $A$ denoted by $D_A(n)$ to be the least natural number $k$ such that for any sequence $(x_1,\dots ,x_k)\in {\mathbb{Z}}^k$, there exist a non-empty subsequence $(x_j,\dots ,x_{j_i})$ and $(a_1,\dots ,a_l)\in A^t$ such that $\sum _{i=1}^n{a}_i{x}_{j_i}\equiv 0(\mathrm{mod}n)$. Das Adhikari and Rath conjectured that for any set $A\subseteq {\mathbb{Z}}_n\setminus \{0\}$, the equality $E_A(n)=D_A(n)+n–1$ holds. In this note, we determine some Davenport constants with weights and also prove that the conjecture holds in some special cases.