Spectral properties of a class of Sierpinski-type Moran measures on $ \mathbb{R}^n $
Let the infinite convolutions \begin{equation*} μ_{\{R_{k}\},\{D_{k}\}}=δ_{R_{1}^{-1}D_{1}}*δ_{R_{1}^{-1}R_{2}^{-1}D_{2}}*δ_{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). \( \{x: \sum_{d\in D_{k}}e^{2πi\langle d,x \rangle}=0\} =\cup_{i=1}^{ϕ(k)}\cup_{j=1}^{m-1}(\frac{j}{m}ν_{k,i}+\mathbb{Z}^{n}) \) for some \( ν_{k,i} \in \{ (l_1, \cdots, l_n)^t : l_i \in [1, m-1] \cap \mathbb{Z}, 1\leq i\leq n \} \). In this paper, we study the spectrality of $μ_{\{R_{k}\},\{D_{k}\}}$, and some necessary and sufficient conditions for \( L^{2}(μ_{\{R_{k}\},\{D_{k}\}}) \) to have an orthogonal exponential function basis are established. Finally, we discuss the explanations and applications of our results.