Scaling of spectra of a class of random convolution on R

Given a Borel probability measure μ on R and a real number p. We call p a spectral eigenvalue of the measure μ if there exists a discrete set Λ such that the setsE(Λ):={e2πiλx:λ∈Λ}andE(pΛ):={e2πipλx:λ∈Λ} are both orthonormal basis for Hilbert space L2(μ). In the present paper, we determine the spectral eigenvalues of a class of random convolution on R.