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 Salem and Sarnak, this proves that for every fixed θ > 1 the spectrum of the convolution operator f → µ θ * f in L p (S 1 ) (where S 1 is the circle group) is countable and is the same for all p ∈ (1, ∞), namely, { µ θ (n) : n ∈ Z}. Our result answers the question raised by P. Sarnak in [8] . We also consider the sets { µ θ

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... : 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 non-empty interval) for Lebesgue-a.e. r > 0.