On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions

We prove that the set of exceptional λ∈(1/2,1)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...

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Autor: Shmerkin, Pablo Sebastian
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2014
País:Argentina
Institución:Consejo Nacional de Investigaciones Científicas y Técnicas
Repositorio:CONICET Digital (CONICET)
Idioma:inglés
OAI Identifier:oai:ri.conicet.gov.ar:11336/35642
Acceso en línea:http://hdl.handle.net/11336/35642
Access Level:acceso abierto
Palabra clave:Bernoulli Convolutions
Self-Similarity
Absolute Continuity
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
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spelling On the Exceptional Set for Absolute Continuity of Bernoulli ConvolutionsShmerkin, Pablo SebastianBernoulli ConvolutionsSelf-SimilarityAbsolute Continuityhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We prove that the set of exceptional λ∈(1/2,1)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.Fil: Shmerkin, Pablo Sebastian. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaSpringer2014-06info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/zipapplication/pdfhttp://hdl.handle.net/11336/35642Shmerkin, Pablo Sebastian; On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions; Springer; Geometric And Functional Analysis; 24; 3; 6-2014; 946-9581016-443X1420-8970CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/http://link.springer.com/article/10.1007%2Fs00039-014-0285-4info:eu-repo/semantics/altIdentifier/doi/10.1007/s00039-014-0285-4info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2024-05-08T14:24:00Zoai:ri.conicet.gov.ar:11336/35642instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982024-05-08 14:24:00.716CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions
title On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions
spellingShingle On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions
Shmerkin, Pablo Sebastian
Bernoulli Convolutions
Self-Similarity
Absolute Continuity
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
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
dc.creator.none.fl_str_mv Shmerkin, Pablo Sebastian
author Shmerkin, Pablo Sebastian
author_facet Shmerkin, Pablo Sebastian
author_role author
dc.subject.none.fl_str_mv Bernoulli Convolutions
Self-Similarity
Absolute Continuity
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
topic Bernoulli Convolutions
Self-Similarity
Absolute Continuity
https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
description We prove that the set of exceptional λ∈(1/2,1)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.
publishDate 2014
dc.date.none.fl_str_mv 2014-06
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/35642
Shmerkin, Pablo Sebastian; On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions; Springer; Geometric And Functional Analysis; 24; 3; 6-2014; 946-958
1016-443X
1420-8970
CONICET Digital
CONICET
url http://hdl.handle.net/11336/35642
identifier_str_mv Shmerkin, Pablo Sebastian; On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions; Springer; Geometric And Functional Analysis; 24; 3; 6-2014; 946-958
1016-443X
1420-8970
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/http://link.springer.com/article/10.1007%2Fs00039-014-0285-4
info:eu-repo/semantics/altIdentifier/doi/10.1007/s00039-014-0285-4
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/zip
application/pdf
dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 15,812455

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