Spectra of Bernoulli convolutions as multipliers in Lp on the circle

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