Extremal sequences for the unit-weighted Gao constant of $\mathbb Z_n$

TL;DR

This paper characterizes extremal sequences related to the Gao constant in cyclic groups, showing that for odd n they are of a standard form, while for even n, non-standard examples exist, with specific results for n=2^rp.

Contribution

It provides a complete characterization of extremal sequences for the unit-weighted Gao constant in $Z_n$, including the standard type and non-standard examples for even n.

Findings

For odd n, all extremal sequences are of the standard type.

For even n, extremal sequences can be non-standard.

Specific characterization for n=2^rp, with p an odd prime.

Abstract

For $A\subseteq \mathbb Z_n$, the $A$-weighted Gao constant $E_A(n)$ is defined to be the smallest natural number $k$, such that any sequence of $k$ elements in $\mathbb Z_n$ has a subsequence of length $n$, whose $A$-weighted sum is zero. Sequences of length $E_A(n)-1$ in $\mathbb Z_n$, which do not have any $A$-weighted zero-sum subsequence of length $n$ are called $A$-extremal sequences for the Gao constant. Such a sequence which has $n-1$ zeroes is said to be of the standard type. When $A=U(n)$ (units in $\mathbb Z_n$) where $n$ is odd, we characterize all such sequences and show that they are of the standard type. When $n$ is even, we give examples of such sequences which are not of the standard type. We also characterize the $U(n)$-extremal sequences for the Gao constant, when $n=2^rp$, where $p$ is an odd prime.