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 $M=\sup \{N_n:n=1,2,3,\dots \}<\infty$. The associated Borel probability measure $${\mu }_{\rho ,\{{\mathcal{D}}_n\}}={\delta }_{\rho {\mathcal{D}}_1}\ast {\delta }_{{\rho }^2{\mathcal{D}}_2}\ast {\delta }_{{\rho }^3{\mathcal{D}}_3}\ast \cdots$$ 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 $n\ge 2$. 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 $L^2({\mu }_{\rho ,\{{\mathcal{D}}_n\}})$.