Extremal sequences related to the Jacobi symbol

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 $\left(\frac{x}{n}\right)$ denote the Jacobi symbol of $x\in {\mathbb{Z}}_n$. We characterize the $C$-extremal sequences for the weight-set $S(n)=\{x\in U(n):(\frac{x}{n})=1\}$ and for the weight-set $L(n;p)=\{x\in U(n):(\frac{x}{n})=(\frac{x}{p}\left)\right\}$ 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.

15 pages. arXiv admin note: substantial text overlap with arXiv:2111.14477