Spectral properties of a class of Sierpinski-type Moran measures on $ \mathbb{R}^n $

TL;DR

This paper investigates the spectral properties of a class of fractal measures generated by infinite convolutions related to Sierpinski-type structures, establishing conditions for the existence of orthogonal exponential bases in their L^2 spaces.

Contribution

It provides new necessary and sufficient conditions for spectrality of these Moran measures, expanding understanding of their harmonic analysis properties.

Findings

Established criteria for spectrality of Moran measures.

Identified conditions for orthogonal exponential bases in L^2 spaces.

Connected spectral properties to geometric and algebraic structures of the measures.

Abstract

Let the infinite convolutions \begin{equation*} \mu_{\{R_{k}\},\{D_{k}\}}=\delta_{R_{1}^{-1}D_{1}}*\delta_{R_{1}^{-1}R_{2}^{-1}D_{2}}*\delta_{R_{1}^{-1}R_{2}^{-1}R_{3}^{-1}D_{3}}*\dotsi \end{equation*} be generated by the sequence of pairs \(\{ (R_k,D_k) \}_{k=1}^{\infty} \), where $ R_k\in M_n(\mathbb{Z})$ is an expanding integer matric, $D_k$ is a finite integer digit sets that satisfies the following two conditions: (i). \( \# D_k = m \) and \( m>2 \) is a prime; (ii). \( \{x: \sum_{d\in D_{k}}e^{2\pi i\langle d,x \rangle}=0\} =\cup_{i=1}^{\phi(k)}\cup_{j=1}^{m-1}(\frac{j}{m}\nu_{k,i}+\mathbb{Z}^{n}) \) for some \( \nu_{k,i} \in \{ (l_1, \cdots, l_n)^t : l_i \in [1, m-1] \cap \mathbb{Z}, 1\leq i\leq n \} \). In this paper, we study the spectrality of $\mu_{\{R_{k}\},\{D_{k}\}}$, and some necessary and sufficient conditions for \( L^{2}(\mu_{\{R_{k}\},\{D_{k}\}}) \) to have an orthogonal exponential function basis are established. Finally, we discuss the explanations and applications of our results.