On unit-weighted zero-sum constants of ${\mathbb{Z}}_n$

Abstract

Given $A\subseteq {\mathbb{Z}}_n$, the constant $C_A(n)$ is defined to be the smallest natural number $k$ such that any sequence of $k$ elements in ${\mathbb{Z}}_n$ has an $A$-weighted zero-sum subsequence having consecutive terms. The value of $C_{U(n)}(n)$ is known when $n$ is odd. We give a different argument to determine the value of $C_{U(n)}(n)$ for any $n$. A $C$-extremal sequence for $U(n)$ is a sequence in ${\mathbb{Z}}_n$ whose length is $C_{U(n)}(n)-1$ and which does not have any $U(n)$-weighted zero-sum subsequence having consecutive terms. We characterize the $C$-extremal sequences for $U(n)$ when $n$ is a power of 2. For any $n$, we determine the value of $C_A(n)$ where $A$ is the set of all odd (or all even) elements of ${\mathbb{Z}}_n$ and also when $A=\{1,2,\dots ,r\}$ where $r<n$.

17 pages