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

TL;DR

This paper establishes a characterization of spectral sets in the p-adic field and related finite rings, linking spectrality to tiling and homogeneity, and constructs examples of singular spectral measures.

Contribution

It proves that in $Q_p$, spectral sets are exactly the tiling sets and are characterized by p-homogeneity, also extending to finite rings and constructing singular spectral measures.

Findings

Spectral sets in $Q_p$ are equivalent to tiling sets.

Spectral sets are characterized by p-homogeneity.

Constructed examples of singular spectral measures, including self-similar ones.

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.