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 $\{(R_k,D_k){\}}_{k=1}^{\mathrm{\infty }}$, 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). $\mathrm{\#}{D}_k=m$ and $m>2$ is a prime; (ii). $\{x:{\sum }_{d\in D_k}{e}^{2\pi i⟨d,x⟩}=0\}={\cup }_{i=1}^{\phi (k)}{\cup }_{j=1}^{m-1}(\frac{j}{m}{\nu }_{k,i}+{\mathbb{Z}}^n)$ for some ${\nu }_{k,i}\in \{(l_1,\cdots ,l_n{)}^t:l_i\in [1,m-1]\cap \mathbb{Z},1\le i\le n\}$. 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.