On Spectral Eigenvalue Problem of a Class of Self-similar Spectral Measures with Consecutive Digits

Abstract

Let $$\mu _{p,q}$$ be a self-similar spectral measure with consecutive digits generated by an iterated function system $$\{f_i(x)=\frac{x}{p}+\frac{i}{q}\}_{i=0}^{q-1}$$ , where $$2\le q\in {{\mathbb {Z}}}$$ and q|p. It is known that for each $$w=w_1w_2\cdots \in \{-1,1\}^\infty :=\{i_1i_2\cdots :~\text {all}~i_k\in \{-1,1\}\}$$ , the set $$\begin{aligned} \Lambda _w=\bigg \{\sum _{j=1}^{n}a_j w_j p^{j-1}:a_j\in \{0,1,\ldots ,q-1\},n\ge 1\bigg \} \end{aligned}$$ is a spectrum of $$\mu _{p,q}$$ . In this paper, we study the possible real number t such that the set $$t\Lambda _w$$ are also spectra of $$\mu _{p,q}$$ for all $$w\in \{-1,1\}^\infty $$ .