Spectra of Bernoulli convolutions as multipliers in Lᵖ on the circle

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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 Salem and Sarnak, this proves that for every fixed θ>1 the spectrum of the convolution operator f μθ*f in Lᵖ(S¹) (where S¹ is the circle group) is countable and is the same for all p∈(1,∞), namely, {μθ(n): n}. Our result answers the question raised by P. Sarnak in Sar. We also consider the sets {μθ(rn): n} 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∈(θ) but uncountable (a non-empty interval) for Lebesgue-a.e. r>0.