It is shown that the closure of the set of Fourier coefficients of the Bernoulli convolution $μ_θ$ 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 in $L^p(S^1)$ (where $S^1$ is the circle group) is countable and is the same for all , namely, . Our result answers the question raised by Sarnak in . We also consider the sets 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 but uncountable (a nonempty interval) for Lebesgue-a.e. $r>0$.
Nikita Sidorov. Boris Solomyak. "Spectra of Bernoulli convolutions as multipliers in $L^p$ on the circle." Duke Math. J. 120 (2) 353 - 370, 1 November 2003. https://doi.org/10.1215/S0012-7094-03-12025-6