Zero-sum constants related to the Jacobi symbol

TL;DR

This paper investigates the $A$-weighted Gao constant in modular arithmetic, determining its value for units and characterizing extremal sequences, especially when the modulus is a power of two.

Contribution

It explicitly computes the $A$-weighted Gao constant for units in $Z_n$ and characterizes extremal sequences for powers of two.

Findings

Calculated $E_A(n)$ for units in $Z_n$

Characterized extremal sequences when $n$ is a power of 2

Determined related constants $C_A(n)$ and $D_A(n)$

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. When $A$ is the set of all units in $\mathbb Z_n$, we determine the value of $E_A(n)$ and values of two related constants $C_A(n)$ and $D_A(n)$. We also characterize all sequences of length $E_A(n)-1$ in $\mathbb Z_n$ which do not have any $A$-weighted zero-sum subsequence of length $n$ when $n$ is a power of 2.