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

Abstract

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 $f\mapsto {\mu }_{\theta }\ast f$ in $L^p(S^1)$ (where $S^1$ is the circle group) is countable and is the same for all $p\in (1,\infty )$, namely, $\overline{\{\widehat{{\mu }_{\theta }}(n):n\in \mathbb{Z}\}}$. Our result answers the question raised by Sarnak in [8]. We also consider the sets $\overline{\{\widehat{{\mu }_{\theta }}(rn):n\in \mathbb{Z}\}}$ 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 $r\in \mathbb{Q}(\theta )$ but uncountable (a nonempty interval) for Lebesgue-a.e. $r>0$.