The Multiple Spectra Of The One-fourth Cantor Measure And Bernoulli Convolutions

In the paper,we mainly discuss the multiple spectra of two classes of famous self-affine measures-the one-fourth Cantor measure and Bernoulli convolutions with two parts. The research on the spectrality of self-affine measures μM,D starts with the one-fourth Cantor measure μM,D(that is, the case M=4, D={0,2}). Based on the previous work for the spectral sets, Jorgensen and Pedersen first observed in1998that the fractal measure μM,D is spectral, moreover, its spectrum A(M,S) has close relations with the compatible pair (M-1D, S), where S={0,1}. The recent studies on this subject illustrate that for some odd integer l, the set lΛ(M, S) is also a spectrum for μM,D.This is rather striking because some spectra for the measure μM,D are thinning. In addition, the other research topic for this paper is Bernoulli convolution μλ associated with λ∈(0,1). The well-known result of Jorgensen and Pedersen shows that if λ=1/2k for some k∈N, then μ1/2k is a spectral measure with spectrum Γ(1/2k). The recent research on the spectrality of μλ shows that μλ is a spectral measure only if λ=1/2k for some k∈N N. Moreover,the spectra of μλ also have sparse property,that is,for some odd integer l, the multiple set lΓ(1/2k) is also a spectrum for μ1/2k.We mainly characterize respectively the number l which has the above property to discuss the multiple spectra of the two self-affine measures-the one-fourth Cantor measure and Bernoulli convolutions. We apply the properties of congruences and the order of elements in the finite group. In the discussion about the one-fourth Cantor measure, three cases on the l, that is, the cases when l is a prime, l is a power of the prime, and l is a product of primes are considered respectively.We obtain several conditions on l such that lΛ(M, S) is a spectrum for μM,D.The result here extends the corresponding result of Dutkay and Jorgensen. And in the discussion about Bernoulli convolutions, we also obtain several conditions on l such that lΓ(1/2k) is a spectrum for μ1/2k.