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

Abstract

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