On the spectrality of a class of Moran measures

TL;DR

This paper investigates the spectral properties of a class of Moran measures, identifying conditions under which they admit orthonormal bases of exponential functions and exploring their tiling properties when absolutely continuous.

Contribution

It establishes criteria for Moran measures to be spectral and characterizes their tiling behavior when the measures are absolutely continuous.

Findings

Existence of a countable spectral set for Moran measures.

Spectrality linked to the measure's absolute continuity.

Support sets of certain measures tile the real line with integers.

Abstract

In this paper, we study the spectrality of a class of Moran measures $\mu_{\mathcal{P},\mathcal{D}}$ on $\mathbb{R}$ generated by $\{(p_n,\mathcal{D}_n)\}_{n=1}^{\infty}$, where $\mathcal{P}=\{p_n\}_{n=1}^{\infty}$ is a sequence of positive integers with $p_n>1$ and $\mathcal{D}=\{\mathcal{D}_{n}\}_{n=1}^{\infty}$ is a sequence of digit sets of $\mathbb{N}$ with the cardinality $\#\mathcal{D}_{n}\in \{2,3,N_{n}\}$. We find a countable set $\Lambda\subset\mathbb{R}$ such that the set $\{e^{-2\pi i \lambda x}|\lambda\in\Lambda\}$ is a orthonormal basis of $L^{2}(\mu_{\mathcal{P},\mathcal{D}})$ under some conditions. As an application, we show that when $\mu_{\mathcal{P},\mathcal{D}}$ is absolutely continuous, $\mu_{\mathcal{P},\mathcal{D}}$ not only is a spectral measure, but also its support set tiles $\mathbb{R}$ with $\mathbb{Z}$.