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 $\mu_\theta$ 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 $p\in(1,\infty)$, 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 $r\in\Q(\theta)$ but uncountable (a non-empty interval) for Lebesgue-a.e. $r>0$.