On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions

Autores
Shmerkin, Pablo Sebastian
Año de publicación
2014
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
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.
Fil: Shmerkin, Pablo Sebastian. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Materia
Bernoulli Convolutions
Self-Similarity
Absolute Continuity
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/35642
Acceder
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network_name_str CONICET Digital (CONICET)
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 Técnicas2025-09-29T10:41:23Zoai: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:34982025-09-29 10:41:23.281CONICET 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
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
topic Bernoulli Convolutions
Self-Similarity
Absolute Continuity
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv 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.
Fil: Shmerkin, Pablo Sebastian. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
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
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
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 13.070432

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