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

Let μM,D be the self-similar measure generated by the positive integer M=RN^q and the product-form digit set D=\{0,1,,N-1\}⊕ Np_1\{0,1,,N-1\}⊕ ⋯ ⊕ Np_s\{0,1,,N-1\}, where R>1, N>1, q, p_i(1≤ i≤ s) are positive integers with gcd(R,N)=1 and p_1<p_2<⋯<p_s<q. In this paper, we first show that μM,D is a spectral measure with a model spectrum Λ. Then we completely settle two types of spectral eigenvalue problems for μM,D. On the first case, for a real t, we give a necessary and sufficient condition under which tΛ is also a spectrum of μM,D. On the second case, we characterize all possible real numbers t such that there exists a countable set Λ'⊂ R such that Λ' and tΛ' are both spectra of μM,D.