2006.378: Spectra of Bernoulli convolutions as multipliers in Lp on the circle

2006.378: Nikita Sidorov and Boris Solomyak (2003) Spectra of Bernoulli convolutions as multipliers in Lp on the circle. Duke Mathematical Journal, 120 (2). pp. 353-370. ISSN 0012-7094

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DOI: 10.1215/S0012-7094-03-12025-6

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.

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