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 $\{(p_n,{\mathcal{D}}_n){\}}_{n=1}^{\mathrm{\infty }}$, where $\mathcal{P}=\{p_n{\}}_{n=1}^{\mathrm{\infty }}$ is a sequence of positive integers with $p_n>1$ and $\mathcal{D}=\{{\mathcal{D}}_n{\}}_{n=1}^{\mathrm{\infty }}$ is a sequence of digit sets of $\mathbb{N}$ with the cardinality $\mathrm{\#}{\mathcal{D}}_n\in \{2,3,N_n\}$. We find a countable set $\mathrm{\Lambda }\subset \mathbb{R}$ such that the set $\{e^{-2\pi i\lambda x}\mathrm{\mid }\lambda \in \mathrm{\Lambda }\}$ 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 $\mathbb{Z}$.