Spectral structure of a class of self-similar spectral measures with product form digit sets

Let \(\mu \) $\mu$ be a Borel probability measure with compact support on \({\mathbb {R}}\) $\mathbb{R}$ , we say \(\mu \) $\mu$ is a spectral measure if there exists a countable set \(\Lambda \subset {\mathbb {R}}\) $\mathrm{\Lambda }\subset \mathbb{R}$ such that the collection of exponential functions \(E(\Lambda ):=\{e^{-2\pi i\langle \lambda , x\rangle }: \lambda \in \Lambda \}\) $E(\mathrm{\Lambda }):=\{e^{-2\pi i⟨\lambda ,x⟩}:\lambda \in \mathrm{\Lambda }\}$ forms an orthonormal basis for the Hilbert space \(L^2(\mu )\) $L^2(\mu )$ . In this case, \(\Lambda \) $\mathrm{\Lambda }$ is called a spectrum of \(\mu \) $\mu$ . In this paper, we first characterize the spectral structure of self-similar spectral measures \(\mu _{t,D}\) ${\mu }_{t,D}$ on \({\mathbb {R}}\) $\mathbb{R}$ , where D is a strict product-form digit set with respect to an integer b and t is an integer which has a proper factor b. And then we settle the spectral eigenvalue (or scaling spectrum) problem for the spectral measure \(\mu _{t,D}\) ${\mu }_{t,D}$ .