On the spectrality of a class of Moran measures

Abstract: 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 $\mathbb{Z}$.