## 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 → μθ * f in L^{p}(S^{1}) (where S ^{1} is the circle group) is countable and is the same for all p ∈ (1, ∞), namely, {μθ(n): n ∈ ℤ}. Our result answers the question raised by Sarnak in [8]. We also consider the sets (μθ(rn): n ∈ 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 > ℚ(θ) but uncountable (a nonempty interval) for Lebesgue-a.e. r > 0.

Original language | English |
---|---|

Pages (from-to) | 353-370 |

Number of pages | 18 |

Journal | Duke Mathematical Journal |

Volume | 120 |

Issue number | 2 |

DOIs | |

State | Published - 1 Nov 2003 |

Externally published | Yes |

### Funding

Funders | Funder number |
---|---|

Directorate for Mathematical and Physical Sciences | 0099814 |

## Fingerprint

Dive into the research topics of 'Spectra of Bernoulli convolutions as multipliers in L^{p}on the circle'. Together they form a unique fingerprint.