\(\mathbb{Q}_p^d\)中紧致开谱集的一个特征
A characterization of compact open spectral sets in $\mathbb{Q}_p^d$
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中文总结 AI 辅助
研究\(\mathbb{Q}_p^d\)中紧致开集\(\Omega\)，证明其通过平移平铺\(\Qp^d\)以及是谱集的充要条件分别是\(C\)通过平移平铺\((\Z/p^n\Z)^d\)以及是谱集，给出了\(\Omega\)相关性质的判定方法。
中文摘要 AI 辅助
设\(\Omega \subset \Qp^d\)为紧致开集，不妨设\(\Omega = \bigsqcup_{c \in C} (c + p^n\Zp^d)\)，其中\(C \subset (\Z/p^n\Z)^d\)且\(n \in \N\)。我们证明\(\Omega\)通过平移平铺\(\Qp^d\)当且仅当\(C\)通过平移平铺\((\Z/p^n\Z)^d\)。此外，\(\Omega\)是\(\Qp^d\)中的谱集当且仅当\(C\)是\((\Z/p^n\Z)^d\)中的谱集。
英文摘要
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 $C \subset (\Z/p^n\Z)^d$ and $n \in\N$. We prove that $Ω$ tiles $\Qp^d$ by translation if and only if $C$ tiles $(\Z/p^n\Z)^d$ by translation. Moreover, $Ω$ is a spectral set in $\Qp^d$ if and only if $C$ is a spectral set in $(\Z/p^n\Z)^d$.
