Doubly-weighted zero-sum constants

TL;DR

This work develops the framework of doubly-weighted zero-sum constants for sequences in $\mathbb Z_n$ and related modules, defining $C_{A,B}(M)$, $D_{A,B}(M)$, and $E_{A,B}(M)$ and establishing fundamental bounds. It provides exact determinations for key weight pairs, notably $(\mathbf 1,\mathbf 1)$, $(\mathbb Z_n',\mathbf 1)$, and $(A,\mathbb Z_n')$, and gives sharp characterizations of extremal sequences in these cases. The analysis reveals how the unit structure of $\mathbb Z_n$ and the presence of zero-divisors influence zero-sum phenomena, with extensions to $(\mathbb Z_n',B)$ and related configurations. The results unify and extend classical zero-sum theory (EGZ-type results, Cauchy–Davenport) within a broad weighted framework and suggest several avenues for future generalizations to triple-weighted setups and more general rings. The findings have implications for combinatorial number theory and algebraic combinatorics by providing precise thresholds and structural descriptions for weighted zero-sum subsequences.

Abstract

Let $A,B\subseteq\mathbb Z_n$ be given and $S=(x_1,\ldots, x_k)$ be a sequence in $\mathbb Z_n$. We say that $S$ is an $(A,B)$-weighted zero-sum sequence if there exist $a_1,\ldots,a_k\in A$ and $b_1,\ldots,b_k\in B$ such that $a_1x_1+\cdots+a_kx_k=0$ and $b_1a_1+\cdots+b_ka_k=0$. 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.

Theorems & Definitions (66)

• Theorem 1.4
• proof
• Remark 1.5
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Remark 2.5