Compact Open Spectral Sets In ${\mathbb{Q}}_p$

Abstract

In this article, we prove that a compact open set in the field ${\mathbb{Q}}_p$ of $p$-adic numbers is a spectral set if and only if it tiles ${\mathbb{Q}}_p$ by translation, and also if and only if it is $p$-homogeneous which is easy to check. We also characterize spectral sets in $\mathbb{Z}/p^n\mathbb{Z}$ ($p\ge 2$ prime, $n\ge 1$ integer) by tiling property and also by homogeneity. Moreover, we construct a class of singular spectral measures in ${\mathbb{Q}}_p$, some of which are self-similar measures.

32 pages, 8 figures. Revised version to appear in Journal of Functional Analysis

References in corpus (2)

• Complex Hadamard matrices and the Spectral Set Conjecture
• Fuglede's conjecture holds in ${\mathbb{Q}}_p$

Cited by in corpus (2)

• Fuglede's conjecture holds in ${\mathbb{Q}}_p$
• Bounded tiles in ${\mathbb{Q}}_p$ are compact open sets