$Q_p$-weighted zero-sum constants

TL;DR

This paper determines new weighted zero-sum constants in finite cyclic groups, specifically for sequences with weights in a subset of the group, advancing understanding of additive combinatorics in modular settings.

Contribution

It explicitly calculates the constants $E_{Q_p, extbf{1}}$, $C_{Q_p, extbf{1}}$, and $D_{Q_p, extbf{1}}$ for weighted zero-sum sequences, extending previous results to new weighted contexts.

Findings

Calculated the exact value of $E_{Q_p, extbf{1}}$ for prime $p$

Established relationships between $E_{Q_p, extbf{1}}$, $C_{Q_p, extbf{1}}$, and $D_{Q_p, extbf{1}}$

Studied weighted zero-sum constants with weights in subsets of $Q_p$

Abstract

A sequence $S=(x_1,\ldots, x_k)$ in $\mathbb Z_p$ is called a $(Q_p,\mathbf 1)$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in Q_p$ such that $a_1x_1+\cdots+a_kx_k=0$ and $a_1+\cdots+a_k=0$. The constant $E_{Q_p,\mathbf 1}$ is defined to be the smallest positive integer $k$ such that every sequence of length $k$ in $\mathbb Z_p$ has a $(Q_p,\mathbf 1)$-weighted zero-sum subsequence of length $p$. We determine the constant $E_{Q_p,\mathbf 1}$ and the related constants $C_{Q_p,\mathbf 1}$ and $D_{Q_p,\mathbf 1}$. We also study some $(Q_p,B)$-weighted zero-sum constants where $B$ is a subset of $Q_p$.