A sufficient and necessary condition for infinite orthogonal sets on some Moran measures

In this work we shall concentrate on fractal-harmonic analysis of a class of Moran measures. Let \(\{M_n\}_{n=1}^{\infty}\) be a sequence of expanding matrix in \(M_2(\mathbb{Z})\) and \(\{D_n\}_{n=1}^{\infty}\) be a sequence of non-collinear integer digit sets satisfying

$$D_n= \left\{\begin{pmatrix}0\\0\end{pmatrix},\begin{pmatrix}\alpha_{n1}\\\alpha_{n2}\end{pmatrix},\begin{pmatrix}\beta_{n1}\\\beta_{n2}\end{pmatrix},\begin{pmatrix}-\alpha_{n1}-\beta_{n1}\\-\alpha_{n2}-\beta_{n2}\end{pmatrix} \right\}.$$

The associated Moran-type measure \(\mu_{\{M_n\},\{D_n\}}\) is generated by the infinite convolution

$$\mu_{\{M_n\},\{D_n\}}=\delta_{M_{1}^{-1}D_1}\ast\delta_{M_{1}^{-1}M_{2}^{-1}D_2}\ast\delta_{M_{1}^{-1}M_{2}^{-1} M_{3}^{-1}D_3}\ast\cdots$$

in the weak\(^*\) -topology. Our result shows that if \(\{\alpha_{n1}\alpha_{n2}\beta_{n1}\beta_{n2}\}_{n=1}^{\infty}\) is bounded, then \(L^{2}(\mu_{\{M_n\},\{D_n\}})\) admits an infinite orthogonal set of exponential functions if and only if there exists a subsequence \(\{n_{k}\}_{k=1}^{\infty}\) of \(\{n_{k}\}_{k=1}^{\infty}\) such that \(\det(M_{n_{k}})\in 2\mathbb{Z}\) .