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 $\alpha \geq \beta$, and $A=\left{x\in\mathbb{N}\; : \; 1 \le a \le p^{\alpha} \; \mbox{ and }\; \gcd(a, p) = 1\right }$, 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.