Spectra of Cantor measures with consecutive digit sets revisited

Abstract

Let ${\mu }_{b,q}$ be the self-similar measure satisfying ${\mu }_{b,q}(\cdot )=\frac{1}{q}{\sum }_{j=0}^{q-1}{\mu }_{b,q}\left({\phi }_j^{-1}\right(\cdot \left)\right)$, where ${\phi }_j\left(x\right)=\frac{x}{b}+\frac{j}{q}$, $0\le j<q$ and $2\le q<b\in Z$ such that $q|b$. This paper will analyze the orthonormal bases of exponential functions for $L^2\left({\mu }_{b,q}\right)$. We present a sufficient and necessary condition for discrete sets to be maximal orthogonal sets and a sufficient condition for maximal orthogonal sets to be bases in $L^2\left({\mu }_{b,q}\right)$ which generalizes the main results of Dutkay, Han and Sun (2009) for ${\mu }_{4,2}$. Finally, a complete characterization on the structure of spectra for ${\mu }_{b,q}$ is given in the viewpoint of measure and dimension which generalizes one of the main results of Deng, Fu and Kang (2024).