Spectral eigenvalue set of self-similar measures associated with product-form Hadamard triples

Previously, An \cite{AL01} showed that the self-similar measure $\mu$ generated by a product-form Hadamard triple is a spectral measure.

In this paper, we study its spectral eigenvalue problem.

A set $A\subset\mathbb R$ is called a spectral eigenvalue set of $\mu$ if there exists a spectrum $\Lambda$ of $\mu$ such that $a\Lambda$ is a spectrum of $\mu$ for every $a\in A$.

We introduce the Product-form Hadamard multiplier set $\mathcal{T}_*$, and prove that for any $s\in [0,\frac{\log \#\mathcal{D}}{\log N}]$, the spectral eigensubspace $$V^{(s)}(\mu_{N,\mathcal{D}},\mathcal{T}_*):=\{\Lambda :t \Lambda \text{ is a spectrum of }\mu \text{ for all }t \in\mathcal{T}_* \text{ and } \dim_{Be}(\Lambda)=s\}$$ has the cardinality of the continuum.

This result allows us to show that for the four-digit self-similar measures, a real number $t$ is a spectral eigenvalue if and only if $t \in \left\{\frac{u}{v}:u,v\in 2\mathbb{Z}+1\right\}$.

And for any subset $S$ of $\mathbb{R}$ is a spectral eigenvalue set if and only if $S \subset t^{-1} (2\mathbb{Z}+1)$ for some $t\in 2\mathbb{Z}+1$.