Square-weighted zero-sum constants

TL;DR

This paper determines the minimal sequence length in modular integers ensuring the existence of a square-weighted zero-sum subsequence, including the case with consecutive terms, advancing understanding of weighted zero-sum problems.

Contribution

It establishes exact bounds for the minimal sequence length guaranteeing square-weighted zero-sum subsequences in ",

Findings

Identifies the minimal length for general zero-sum subsequences.

Determines the minimal length for consecutive zero-sum subsequences.

Provides explicit formulas depending on the structure of "

Abstract

Let $A\subseteq \mathbb Z_n$ be a subset. A sequence $S=(x_1,\ldots,x_k)$ in $\mathbb Z_n$ is said to be an $A$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in A$ such that $a_1x_1+\cdots+a_kx_k=0$. By a square, we shall mean a non-zero square in $\mathbb Z_n$. We determine the smallest natural number $k$, such that every sequence in $\mathbb Z_n$ whose length is $k$, has a square-weighted zero-sum subsequence. We also determine the smallest natural number $k$, such that every sequence in $\mathbb Z_n$ whose length is $k$, has a square-weighted zero-sum subsequence whose terms are consecutive terms of the given sequence.