Extremal sequences related to the Jacobi symbol

TL;DR

This paper characterizes extremal sequences in modular arithmetic related to the Jacobi symbol, identifying sequences that avoid weighted zero-sum subsequences for specific sets based on quadratic residues.

Contribution

It provides a complete characterization of $C$-extremal and $D$-extremal sequences for weight-sets defined via the Jacobi symbol, extending zero-sum theory in number theory.

Findings

Characterized $C$-extremal sequences for sets based on quadratic residues.

Extended the concept to $D$-extremal sequences for similar weight-sets.

Connected extremal sequences to properties of the Jacobi symbol and modular units.

Abstract

For a weight-set $A\subseteq \mathbb Z_n$, the $A$-weighted zero-sum 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 of consecutive terms. A sequence of length $C_A(n)-1$ in $\mathbb Z_n$ which does not have any $A$-weighted zero-sum subsequence of consecutive terms will be called a $C$-extremal sequence for $A$. Let $\big(\frac{x}{n}\big)$ denote the Jacobi symbol of $x\in\mathbb Z_n$. We characterize the $C$-extremal sequences for the weight-set $S(n)=\big\{\,x\in U(n):\big(\frac{x}{n}\big)=1\,\big\}$ and for the weight-set $L(n;p)=\big\{\,x\in U(n):\big(\frac{x}{n}\big)=\big(\frac{x}{p}\big)\,\big\}$ where $p$ is a prime divisor of $n$. We can define $D$-extremal sequences for these weight-sets in a way analogous to the definition of $C$-extremal sequences. We also characterize these sequences.