Spectrality of Some One-Dimensional Moran Measures

Abstract

A Borel probability measure $\mu$ with compact support on ${\mathbb {R}}^d$ is called spectral measure if there exists a discrete set $\Lambda \subset {\mathbb {R}}^d$ such that $E_\Lambda :=\{e^{2\pi i\langle \lambda ,x\rangle }:\lambda \in \Lambda \}$ forms an orthonormal basis of $L^2(\mu )$. In this paper, we first study the spectrality of a class of general Moran measures on ${\mathbb {R}}$. Suppose that $p_n\ge 2$ and $\{(p_n, B_n, L_n)\}_{n=1}^{\infty }$ is a sequence of Hadamard triples. We show that if $0\in B_n, \gcd B_n=1$ for $n\ge 1$ and $\sup _{n\ge 1}\{\sup _{b\in B_n}|b|\}<\infty$, then the associated Moran measure $\mu _{\{p_n,B_n\}}=\delta _{p_1^{-1}B_1}*\delta _{(p_1p_2)^{-1}B_2}*\cdots$ is a spectral measure. Secondly, we use the above result to deal with the Moran measure generated by $p_n\ge 2$ and ${\mathcal {D}}_n=\{0,a_n,b_n\}$ with $\gcd (a_n, b_n)=1$. We prove that if $\sup _{n\ge 1}\{|a_n|/p_n,\; |b_n|/p_n\}<\infty$, then $\mu _{\{p_n,{\mathcal {D}}_n\}}$ is a spectral measure if and only if $\{a_n,b_n\}=\{\pm 1\}\pmod 3$ for $n\ge 1$ and $3\mid p_n$ for $n\ge 2$.