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 $$K(q,A)=\{\sum _{i=1}^{\mathrm{\infty }}\frac{a_i}{q^i}:a_i\in A\mathrm{\forall }i\in \mathbb{N}\}\mathrm{.}$$ 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 $E(\mathrm{\Lambda })=\{e^{2\pi \mathrm{i}\lambda x}:\lambda \in \mathrm{\Lambda }\}$ and $E(t\mathrm{\Lambda })=\{e^{2\pi \mathrm{i}t\lambda x}:\lambda \in \mathrm{\Lambda }\}$ are orthonormal bases in $L^2(\mu )$ for some $\mathrm{\Lambda }\subset \mathbb{R}$. For any self-similar spectral measure generated by a Hadamard triple, we provide a class of spectral eigenvalues which is dense in $[0,+\mathrm{\infty })$, and show that every eigen-subspace associated with these spectral eigenvalues is infinite.