$\{\pm 1\}$-weighted zero-sum constants

TL;DR

This paper investigates the properties of weighted zero-sum sequences in modular arithmetic, specifically determining key constants for the case where weights are ±1 and 1, revealing new combinatorial number theory results.

Contribution

The paper explicitly computes the constants $E_{A,B}(n)$, $C_{A,B}(n)$, and $D_{A,B}(n)$ for the case $A=\{\\pm 1\\}$ and $B=\{1\}$, advancing understanding of weighted zero-sum problems.

Findings

Determined $E_{A,B}(n)$ for $A=\{\\pm 1\\}$, $B=\{1\}$.

Established relations between $E_{A,B}(n)$, $C_{A,B}(n)$, and $D_{A,B}(n)$.

Provided explicit formulas for these constants in the specified case.

Abstract

Let $A,B\subseteq \mathbb Z_n\setminus\{0\}$. A sequence $S=(x_1,\ldots, x_k)$ in $\mathbb Z_n$ is called an $(A,B)$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in A$ and $b_1,\ldots,b_k\in B$ such that $a_1x_1+\cdots+a_kx_k=0$ and $b_1a_1+\cdots+b_ka_k=0$. The constant $E_{A,B}(n)$ is defined to be the smallest positive integer $k$ such that every sequence of length $k$ in $\mathbb Z_n$ has an $(A,B)$-weighted zero-sum subsequence of length $n$. We determine the constant $E_{A,B}(n)$ and the related constants $C_{A,B}(n)$ and $D_{A,B}(n)$ when $A=\{\pm 1\}$ and $B=\{1\}$.