Rational points in Cantor sets and spectral eigenvalue problem for self-similar spectral measures
Derong Kong, Kun Li, Zhiqiang Wang
TL;DR
The paper links arithmetic structure of rational points in Cantor-type self-similar sets to spectral properties of self-similar measures. It proves a sharp equivalence: for gcd$(p,q)=1$, the finiteness of $D_p\cap K(q,A)$ is equivalent to $K(q,A)$ having Hausdorff dimension strictly less than 1 (and hence no interior), and it derives a uniform bound on shifted intersections under suitable digit sets. In the spectral-measure part, it shows that for Hadamard triples with a deficiency ($\#B< N$), a dense family of spectral eigenvalues exists, with corresponding eigen-subspaces infinite, via constructing scaled spectra $p_1^{n_1}\cdots p_k^{n_k}\Lambda$ and identifying the eigenvalues in $\\mathcal{E}_N$. The results advance understanding of rational points in fractal sets and reveal rich, dense structures of spectral eigenvalues for self-similar measures, with implications for spectral theory on fractals and tiling. The methods combine weak separation, periodic codings, order bounds, and Hadamard-triple spectral constructions to yield concrete finiteness and spectral phenomena.
Abstract
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 $μ$ on $\mathbb{R}$, a real number $t\in \mathbb{R}$ is called a spectral eigenvalue of $μ$ if both $E(Λ) =\big\{ e^{2 π\mathrm{i} λx}: λ\in Λ\big\}$ and $E(tΛ) = \big\{ e^{2 π\mathrm{i} tλx}: λ\in Λ\big\}$ are orthonormal bases in $L^2(μ)$ for some $Λ\subset \mathbb{R}$. 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.