Weighted EGZ Constant for p-groups of rank 2

Abstract

Let $G$ be a finite abelian group of exponent $n$, written additively, and let $A$ be a subset of $\mathbb{Z}$. The constant $s_A(G)$ is defined as the smallest integer $\ell$ such that any sequence over $G$ of length at least $\ell$ has an $A$-weighted zero-sum of length $n$ and ${\eta }_A(G)$ defined as the smallest integer $\ell$ such that any sequence over $G$ of length at least $\ell$ has an $A$-weighted zero-sum of length at most $n$. Here we prove that, for $\alpha \ge \beta$, and $A=\left\{x\in\mathbb{N}\; : \; 1 \le a \le p^α \; \mbox{ and }\; \gcd(a, p) = 1\right \}$, we have $s_A({\mathbb{Z}}_{p^{\alpha }}\oplus {\mathbb{Z}}_{p^{\beta }})={\eta }_A({\mathbb{Z}}_{p^{\alpha }}\oplus {\mathbb{Z}}_{p^{\beta }})+p^{\alpha }-1=p^{\alpha }+\alpha +\beta$ and classify all the extremal $A$-weighted zero-sum free sequences.