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, Peres...

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Autor principal: Shmerkin, P.
Formato: JOUR
Materias:
28A80
Bernoulli convolutions
hausdorff dimension
Primary 28A78
Secondary 37A45
self-similar measures
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_1016443X_v24_n3_p946_Shmerkin
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id todo:paper_1016443X_v24_n3_p946_Shmerkin
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spelling todo:paper_1016443X_v24_n3_p946_Shmerkin2023-10-03T15:56:22Z On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions Shmerkin, P. 28A80 Bernoulli convolutions hausdorff dimension Primary 28A78 Secondary 37A45 self-similar measures 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. © 2014 Springer Basel. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_1016443X_v24_n3_p946_Shmerkin
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic 28A80
Bernoulli convolutions
hausdorff dimension
Primary 28A78
Secondary 37A45
self-similar measures
spellingShingle 28A80
Bernoulli convolutions
hausdorff dimension
Primary 28A78
Secondary 37A45
self-similar measures
Shmerkin, P.
On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions
topic_facet 28A80
Bernoulli convolutions
hausdorff dimension
Primary 28A78
Secondary 37A45
self-similar measures
description 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. © 2014 Springer Basel.
format JOUR
author Shmerkin, P.
author_facet Shmerkin, P.
author_sort Shmerkin, P.
title On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions
title_short On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions
title_full On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions
title_fullStr On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions
title_full_unstemmed On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions
title_sort on the exceptional set for absolute continuity of bernoulli convolutions
url http://hdl.handle.net/20.500.12110/paper_1016443X_v24_n3_p946_Shmerkin
work_keys_str_mv AT shmerkinp ontheexceptionalsetforabsolutecontinuityofbernoulliconvolutions
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