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

Let \(\mu \) be a Borel probability measure with compact support on \({\mathbb {R}}\), we say \(\mu \) is a spectral measure if there exists a countable set \(\Lambda \subset {\mathbb {R}}\) such that the collection of exponential functions \(E(\Lambda ):=\{e^{-2\pi i\langle \lambda , x\rangle }: \lambda \in \Lambda \}\) forms an orthonormal basis for the Hilbert space \(L^2(\mu )\). In this case, \(\Lambda \) is called a spectrum of \(\mu \). In this paper, we first characterize the spectral structure of self-similar spectral measures \(\mu _{t,D}\) on \({\mathbb {R}}\), 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 \(\mu _{t,D}\).