Square-weighted zero-sum constants

Abstract

Let $A\subseteq {\mathbb{Z}}_n$ be a subset. A sequence $S=(x_1,\dots ,x_k)$ in ${\mathbb{Z}}_n$ is said to be an $A$-weighted zero-sum sequence if there exist $a_1,\dots ,a_k\in A$ such that $a_1{x}_1+\cdots +a_k{x}_k=0$. 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.

15 pages