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

Let $\mathrm{\Omega }\subset {\text{Qp}}^d$ be a compact open set. Such a set, without lose of generality, admits a representation $\mathrm{\Omega }={⨆}_{c\in C}(c+p^n{\text{Zp}}^d)$, where $C\subset (\mathbb{Z}\mathrm{/}{p}^n\mathbb{Z}{)}^d$ and $n\in \mathbb{N}$. We prove that $\mathrm{\Omega }$ tiles ${\text{Qp}}^d$ by translation if and only if $C$ tiles $(\mathbb{Z}\mathrm{/}{p}^n\mathbb{Z}{)}^d$ by translation. Moreover, $\mathrm{\Omega }$ is a spectral set in ${\text{Qp}}^d$ if and only if $C$ is a spectral set in $(\mathbb{Z}\mathrm{/}{p}^n\mathbb{Z}{)}^d$.