Spectrality of a class of moran measures on $\mathbb{R}^2$

Abstract: Let $\mu_{{M_n},{D_n}}$ be a Moran measure on $\mathbb{R}^2$ generated by a sequence of expanding matrices ${M_n}\subset GL(2, \mathbb{Z})$ and a sequence of integer digit sets ${D_n}$ where $D_n=\left{\begin{pmatrix} 0 \ 0 \end{pmatrix},\begin{pmatrix} \alpha_{n_1} \ \alpha_{n_2} \end{pmatrix},\begin{pmatrix} \beta_{n_1} \ \beta_{n_2} \end{pmatrix},\begin{pmatrix} -\alpha_{n_1}-\beta_{n_1} \ -\alpha_{n_2}-\beta_{n_2} \end{pmatrix} \right}$ with $\alpha_{n_1}\beta_{n_2}-\alpha_{n_2}\beta_{n_1}\notin2\mathbb{Z}$. If $|\det(M_n)|>4$ for $n\geq1$, $\sup\limits_{n\ge 1}\Vert M_n^{-1}\Vert<1$ and $#{D_n: n\ge 1}<\infty$, then we show that $\mu_{{M_n},{D_n}}$ is a spectral measure if and only if $M_n\in GL(2, 2\mathbb{Z})$ for $n\geq2$. If $|\det(M_n)| =4$ for $n\geq1$, we also establish a necessary and sufficient condition for a class of special Moran measures to be spectral measures.