The spectral eigenvalues of a class of product-form self-similar spectral measure

Let ₌, ₃ be the self-similar measure generated by the positive integer M=RNq and the product-form digit set D=\0, 1, , N-1\ N^p₁\0, 1, , N-1\ N^pₛ\0, 1, , N-1\, where R>1, N>1, q, pᵢ (1 i s) are positive integers with gcd (R, N) =1 and p₁<p₂<<pₛ<q. In this paper, we first show that ₌, ₃ is a spectral measure with a model spectrum. Then we completely settle two types of spectral eigenvalue problems for ₌, ₃. On the first case, for a real t, we give a necessary and sufficient condition under which t is also a spectrum of ₌, ₃. On the second case, we characterize all possible real numbers t such that there exists a countable set ' R such that ' and t' are both spectra of ₌, ₃.