On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions

We prove that the set of exceptional λ ∈ ( 1 / 2 , 1 ) such that the associated Bernoulli convolution is singular has zero Hausdorff dimension, and likewise for biased Bernoulli convolutions, with the exceptional set independent of the bias. This improves previous results by Erdös, Kahane, Solomyak,...

Full description

Saved in:
Bibliographic Details
Published inGeometric and functional analysis Vol. 24; no. 3; pp. 946 - 958
Main Author Shmerkin, Pablo
Format Journal Article
LanguageEnglish
Published Basel Springer Basel 01.06.2014
Subjects
Analysis
Mathematics
Mathematics and Statistics
28A80
Bernoulli convolutions
Primary 28A78
self-similar measures
Secondary 37A45
hausdorff dimension
Online AccessGet full text

Cover

More Information
Summary:We prove that the set of exceptional λ ∈ ( 1 / 2 , 1 ) such that the associated Bernoulli convolution is singular has zero Hausdorff dimension, and likewise for biased Bernoulli convolutions, with the exceptional set independent of the bias. This improves previous results by Erdös, Kahane, Solomyak, Peres and Schlag, and Hochman. A theorem of this kind is also obtained for convolutions of homogeneous self-similar measures. The proofs are very short, and rely on old and new results on the dimensions of self-similar measures and their convolutions, and the decay of their Fourier transform.
ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-014-0285-4