On the spectrality of a class of Moran measures

In this paper, we study the spectrality of a class of Moran measures $\mu_{\mathcal{P},\mathcal{D}}$ on $\mathbb{R}$ generated by $\{(p_n,\mathcal{D}_n)\}_{n=1}^{\infty}$, where $\mathcal{P}=\{p_n\}_{n=1}^{\infty}$ is a sequence of positive integers with $p_n>1$ and $\mathcal{D}=\{\mathcal{D}_{n}\}_{n=1}^{\infty}$ is a sequence of digit sets of $\mathbb{N}$ with the cardinality $\#\mathcal{D}_{n}\in \{2,3,N_{n}\}$. We find a countable set $\Lambda\subset\mathbb{R}$ such that the set $\{e^{-2\pi i \lambda x}|\lambda\in\Lambda\}$ is a orthonormal basis of $L^{2}(\mu_{\mathcal{P},\mathcal{D}})$ under some conditions. As an application, we show that when $\mu_{\mathcal{P},\mathcal{D}}$ is absolutely continuous, $\mu_{\mathcal{P},\mathcal{D}}$ not only is a spectral measure, but also its support set tiles $\mathbb{R}$ with $\mathbb{Z}$.