Doubly-weighted zero-sum constants

Abstract

Let $A,B\subseteq {\mathbb{Z}}_n$ be given and $S=(x_1,\dots ,x_k)$ be a sequence in ${\mathbb{Z}}_n$. We say that $S$ is an $(A,B)$-weighted zero-sum sequence if there exist $a_1,\dots ,a_k\in A$ and $b_1,\dots ,b_k\in B$ such that $a_1{x}_1+\cdots +a_k{x}_k=0$ and $b_1{a}_1+\cdots +b_k{a}_k=0$. We show that if $S$ has length $2n-1$, then $S$ has an $(A,B)$-weighted zero-sum subsequence of length $n$. The constant $E_{A,B}$ 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 $n$. A sequence in ${\mathbb{Z}}_n$ of length $E_{A,B}-1$ which does not have any $(A,B)$-weighted zero-sum subsequence of length $n$ is called an $E$-extremal sequence for $(A,B)$. We determine the constant $E_{A,B}$ and characterize the $E$-extremal sequences for some pairs $(A,B)$. We also study the related constants $C_{A,B}$ and $D_{A,B}$ which are defined in the article.

18 pages