The structure of tiles in $\mathbb{Z}_{p^n}\times \mathbb{Z}_q$ and $\mathbb{Z}_{p^n}\times \mathbb{Z}_p$

Abstract

In this paper, we provide a geometric characterization of tiles in the finite abelian groups \( \mathbb{Z}_{p^n} \times \mathbb{Z}_q \) and \( \mathbb{Z}_{p^n} \times \mathbb{Z}_p \) using the concept of a \( p \)-homogeneous tree, which provides an intuitively visualizable criterion.