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

Abstract

Given $q\in {\mathbb{N}}_{\ge 3}$ and a finite set $A\subset \mathbb{Q}$, let $$K(q,A)=\{\sum _{i=1}^{\infty }\frac{a_i}{q^i}:a_i\in A\forall i\in \mathbb{N}\}.$$ 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 $\hat{I}¼$ on $\mathbb{R}$, a real number $t\in \mathbb{R}$ is called a spectral eigenvalue of $\hat{I}¼$ if both $E(\hat{I}›)=\{e^{2\ddot{I}€\mathrm{i}\hat{I}»x}:\hat{I}»\in \hat{I}›\}$ and $E(t\hat{I}›)=\{e^{2\ddot{I}€\mathrm{i}t\hat{I}»x}:\hat{I}»\in \hat{I}›\}$ are orthonormal bases in $L^2(\hat{I}¼)$ for some $\hat{I}›\subset \mathbb{R}$. 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.

16 pages