# Spectra of Bernoulli convolutions as multipliers in Lp on the circle

Sidorov, Nikita and Solomyak, Boris (2003) Spectra of Bernoulli convolutions as multipliers in Lp on the circle. Duke Mathematical Journal, 120 (2). pp. 353-370. ISSN 0012-7094

## Abstract

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 R. Salem and P. Sarnak, this proves that for every fixed θ>1 the spectrum of the convolution operator $f\mapsto \mu\sb \theta\ast f$ in Lp(S1) (where S1 is the circle group) is countable and is the same for all $p\in (1, \infty)$, namely, $\overline{\{\widehat {\mu\sb \theta}(n) : n\in \mathbb {Z}\}}. Our result answers the question raised by Sarnak in [8]. We also consider the sets$\overline{\{\widehat {\mu\sb \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.

Item Type: Article MSC 2010, the AMS's Mathematics Subject Classification > 47 Operator theory Ms Lucy van Russelt 26 Sep 2006 20 Oct 2017 14:12 http://eprints.maths.manchester.ac.uk/id/eprint/614