Spectra of Bernoulli convolutions as multipliers in L^p on the circle

It is shown that the closure of the set of Fourier coefficients of the Bernoulli convolution μθ parameterized by a Pisot number θ is countable. Combined with results of R. Salem and P. Sarnak, this proves that for every fixed θ > 1 the spectrum of the convolution operator f → μθ * f in L^p(S¹) (where S ¹ is the circle group) is countable and is the same for all p ∈ (1, ∞), namely, {μθ(n): n ∈ ℤ}. Our result answers the question raised by Sarnak in [8]. We also consider the sets (μθ(rn): n ∈ ℤ} for r > 0 which correspond to a linear change of variable for the measure. We show that such a set is still countable for all r > ℚ(θ) but uncountable (a nonempty interval) for Lebesgue-a.e. r > 0.