On unit-weighted zero-sum constants of $\mathbb Z_n$

TL;DR

This paper investigates the minimal sequence length in cyclic groups ensuring the existence of weighted zero-sum subsequences with consecutive terms, providing new proofs and characterizations for specific cases including powers of two and particular subsets.

Contribution

It offers a new proof for the value of $C_{U(n)}(n)$ for all $n$, characterizes extremal sequences for powers of two, and determines constants for specific subsets of $Z_n$.

Findings

Determined $C_{U(n)}(n)$ for all $n$ using a new argument.

Characterized $C$-extremal sequences when $n$ is a power of 2.

Calculated $C_A(n)$ for sets of all odd/even elements and initial segments of integers.

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 $A=\{1,2,\ldots,r\}$ where $r<n$.