Spectra of Bernoulli convolutions as multipliers in $L^p$ 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 Salem and Sarnak, this proves that for every fixed $\theta >1$ the spectrum of the convolution operator $f\mapsto \mu_\theta*f$in$L^p(S^1)$(where$S^1$ is the circle group) is countable and is the same for all $p\in (1,\mathrm{\infty })$, namely, $\bar{{\hat{\mu_\theta}(n) : n\in$\mathbb{Z}$}}$. Our result answers the question raised by P. Sarnak in \cite{Sar}. We also consider the sets $\bar{{\hat{\mu_\theta}(rn) : n\in$\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 $r\in \text{Q}(\theta )$ but uncountable (a non-empty interval) for Lebesgue-a.e. $r>0$.