Multiple Spectra of Bernoulli Convolutions

Abstract

Let μ _λ be the Bernoulli convolution associated with λ ∈ (0, 1). The well-known result of Jorgensen and Pedersen shows that if λ = 1/(2k) for some k ∈ ℕ, then μ ₁/(2k) is a spectral measure with spectrum Γ(1/(2k)). The recent research on the spectrality of μ _λ shows that μ _λ is a spectral measure only if λ = 1/(2k) for some k ∈ ℕ. Moreover, for certain odd integer p, the multiple set pΓ(1/(2k)) is also a spectrum for μ ₁/(2k). This is surprising because some spectra for the measure μ ₁/(2k) are thinning. In this paper we mainly characterize the number p that has the above property. By applying the properties of congruences and the order of elements in the finite group, we obtain several conditions on p such that pΓ(1/(2k)) is a spectrum for μ ₁/(2k).