Non-spectrality of Moran measures with consecutive digits

TL;DR

This paper investigates the spectral properties of Moran measures with consecutive digits, establishing conditions under which these measures admit infinite or finite orthogonal exponential sets, and demonstrating non-spectrality in certain cases.

Contribution

It provides new criteria for the existence and limitations of orthogonal exponential functions in Moran measures with consecutive digits, extending previous spectral measure results.

Findings

If $L^2$ contains an infinite orthogonal exponential set, then infinitely many $N_{n}$ share common factors with $q$.

When $(q,N_{n})=1$ and $(p,N_{n})=1$, at most $M$ orthogonal exponentials exist, and this bound is optimal.

If $(q,N_{n})=1$ and $(p,N_{n})>1$, then infinitely many orthogonal exponential functions can exist.

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 $L^{2}(\mu_{\rho, \{\mathcal{D}_{n}\}})$.