Non-spectrality of Moran measures with consecutive digits

Abstract: Let $\rho=(\frac{p}{q})^{\frac{1}{r}}<1$ for some $p,q,r\in\mathbb{N}$ with $(p,q)=1$ and $\mathcal{D}{n}={0,1,\cdot\cdot\cdot,N{n}-1}$, where $N_{n}$ is prime for all $n\in\mathbb{N}$, and denote $M=\sup{N_{n}:n=1,2,3,\ldots}<\infty$. The associated Borel probability measure $$\mu_{\rho,{\mathcal{D}{n}}}=\delta{\rho\mathcal{D}{1}}*\delta{\rho^{2}\mathcal{D}{2}}*\delta{\rho^{3}\mathcal{D}_{3}}*\cdots$$ is called a Moran measure. Recently, Deng and Li proved that $\mu_{\rho,{\mathcal{D}{n}}}$ is a spectral measure if and only if $\frac{1}{N{n}\rho}$ is an integer for all $n\geq 2$. In this paper, we prove that if $L^{2}(\mu_{\rho, {\mathcal{D}{n}}})$ contains an infinite orthogonal exponential set, then there exist infinite positive integers $n{l}$ such that $(q,N_{n_{l}})>1$. Contrastly, if $(q,N_{n})=1$ and $(p,N_{n})=1$ for all $n\in\mathbb{N}$, then there are at most $M$ mutually orthogonal exponential functions in $L^{2}(\mu_{\rho, {\mathcal{D}{n}}})$ and $M$ is the best possible. If $(q,N{n})=1$ and $(p,N_{n})>1$ for all $n\in\mathbb{N}$, then there are any number of orthogonal exponential functions in $L^{2}(\mu_{\rho, {\mathcal{D}_{n}}})$.