A characterization of compact open spectral sets in $\mathbb{Q}_p^d$

Let $\Omega \subset \Qp^d$ be a compact open set.

Such a set, without lose of generality, admits a representation \(\Omega = \bigsqcup_{c \in C} (c + p^n\Zp^d)\), where $C \subset (\Z/p^n\Z)^d$ and $n \in\N$.

We prove that $\Omega$ tiles $\Qp^d$ by translation if and only if $C$ tiles $(\Z/p^n\Z)^d$ by translation.

Moreover, $\Omega$ is a spectral set in $\Qp^d$ if and only if $C$ is a spectral set in $(\Z/p^n\Z)^d$.