Rational points in Cantor sets and spectral eigenvalue problem for self-similar spectral measures

Given $q\in \mathbb{N}_{\ge 3}$ and a finite set $A\subset\mathbb{Q}$, let $$K(q,A)= \bigg\{\sum_{i=1}^{\infty} \frac{a_i}{q^{i}}:a_i \in A ~\forall i\in \mathbb{N} \bigg\}.$$ For $p\in\mathbb{N}_{\ge 2}$ let $D_p\subset\mathbb{R}$ be the set of all rational numbers having a finite $p$-ary expansion. We show in this paper that for $p \in \mathbb{N}_{\ge 2}$ with $\gcd(p,q)=1$, the intersection $D_p\cap K(q, A)$ is a finite set if and only if $\dim_H K(q, A)<1$, which is also equivalent to the fact that the set $K(q, A)$ has no interiors. We apply this result to study the spectral eigenvalue problem. For a Borel probability measure $\mu$ on $\mathbb{R}$, a real number $t\in \mathbb{R}$ is called a spectral eigenvalue of $\mu$ if both $E(\Lambda) =\big\{ e^{2 \pi \mathrm{i} \lambda x}: \lambda \in \Lambda \big\}$ and $E(t\Lambda) = \big\{ e^{2 \pi \mathrm{i} t\lambda x}: \lambda \in \Lambda \big\}$ are orthonormal bases in $L^2(\mu)$ for some $\Lambda \subset \mathbb{R}$. For any self-similar spectral measure generated by a Hadamard triple, we provide a class of spectral eigenvalues which is dense in $[0,+\infty)$, and show that every eigen-subspace associated with these spectral eigenvalues is infinite.