Spectrality of a class of Moran measures on the plane

Let \(\{(R_k,D_k)\}_{k=1}^\infty\) be a sequence of pairs, where

$$D_k=\{0,1,\ldots,q_k-1\}(1,1)^T$$

is an integer vector set and \(R_k\) is an integer diagonal matrix or upper triangular matrix, i.e., \(R_k={\begin{pmatrix} s_k & 0\\ 0 & t_k \end{pmatrix}}\) or \(R_k={\begin{pmatrix} u_k & 1\\ 0 & v_k \end{pmatrix}}\). Associated with the sequence \(\{(R_k,D_k)\}_{k=1}^\infty\) , Moran measure \(\mu_{\{R_k\},\{D_k\}}\) is defined by

$$\mu_{\{R_k\},\{D_k\}}=\delta_{R_{1}^{-1}D_{1}}\ast\delta_{R_{1}^{-1}R_{2}^{-1}D_{2}}\ast\cdots\ast \delta_{R_{1}^{-1}R_{2}^{-1}\cdots R_{k}^{-1}D_{k}}\ast \cdots.$$

In this paper, we consider the spectrality of \(\mu_{\{R_k\},\{D_k\}}\). We prove that \(\mu_{\{R_k\},\{D_k\}}\) is a spectral measure under certain conditions in terms of \((R_k,D_k)\), i.e., there exists a Fourier basis for \(L^2(\mu_{\{R_k\},\{D_k\}})\).