On the Fourier orthonormal bases of a class of Sierpinski-type Moran measures on $ \mathbb{R}^n $

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 ${ (M_k,D_k) }{k=1}^{\infty} $, where $ M_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). ${x: \sum{d\in D_{k}}e^{2\pi i\langle d,x \rangle}=0} =\cup_{i=1}^{\phi(k)}\cup_{j=1}^{m-1}\left(\frac{j}{m}\nu_{k,i}+\mathbb{Z}^{n}\right)$ for some $\nu_{k,i} \in { (l_1, \cdots, l_n)^* : l_i \in [1, m-1] \cap \mathbb{Z}, 1\leq i\leq n }$. 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 demonstrate applications of these results to Moran measures.