Spectrality of infinite Bernoulli convolutions

Let {bk−ak}k=1∞ be a sequence of positive integers with upper bound and let δEδE be the uniformly discrete probability measure on the finite set E  . For 0<ρ<10<ρ<1, the infinite convolutionμρ,{ak,bk}=δρ{a1,b1}⁎δρ2{a2,b2}⁎⋯⋯ is called an infinite Bernoulli convolution. In this paper, we investigate whenever there exists Λ such that {e−2πiλx}λ∈Λ{e−2πiλx}λ∈Λ is an orthonormal basis for L2(μρ,{ak,bk}).