Common Spectral Eigenvalue Spectra for Random Convolutions Generated by Hadamard Triples

Abstract

Lu proved that the set $$\mathcal{T}:=\{t\in\mathbb{Z}\setminus\{0\}:(q,\mathcal{D},tL)\text{ forms a Hadamard triple}\}$$ constitutes a spectral eigenvalue set for $\mu_{q,D}$, where $(q,\mathcal{D},L)$ is a Hadamard triple. And they prove for $s \in [0,\frac{\log \#\mathcal{D}}{\log q}]$, the corresponding spectra form a family of cardinality continuum. In this paper, we study Moran measures formed by random convolutions of finite Hadamard triples. Let $$\mu=\delta_{M_1^{-1}D_1}*\delta_{M_2^{-1}D_2}*\cdots, \qquad M_k=q_1q_2\cdots q_k,$$ where the factors are produced from finitely many triples $\{(N_j,B_j,L_j):1\le j\le m\}$, $(\omega_k)_{k=1}^{\infty}\in\{1,2,\ldots,m\}^{\mathbb N}$, and $n_k\in\mathbb N^+$, by setting $$ q_k=N_{\omega_k}^{n_k},\qquad D_k=B_{\omega_k},\qquad E_k=N_{\omega_k}^{n_k-1}L_{\omega_k}. $$ Assume a non-full-digit gap $$\rho:=\min_{1\le j\le m}\frac{N_j}{\#B_j}>1.$$ For the common Hadamard triple multiplier set $$\mathcal{T}_*:=\bigcap_{j=1}^m\{t\in\mathbb{Z}\setminus\{0\}:(N_j,B_j,tL_j)\text{ is a Hadamard triple}\},$$ Our main result is that, for every $$0\le s\le \kappa_\omega:=\limsup_{R\to\infty}\frac{\sum_{r=1}^R\log \#D_r}{\sum_{r=1}^R\log q_r},$$ there exist continuum many countable sets $\Lambda\subset\mathbb{Z}$ such that $t\Lambda$ is a spectrum of $\mu$ for every $t\in\mathcal{T}_*$ and $\dim_{Be}\Lambda=s$.