On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions

Autores
Shmerkin, Pablo Sebastian
Año de publicación
2019
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the Lq dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg's long-standing conjecture on the dimension of the intersections of ×p- and ×q-invariant sets. Among several other applications, we also show that Bernoulli convolutions have an Lq density for all finite q, outside of a zero-dimensional set of exceptions. The proof of the main result is inspired by M. Hochman's approach to the dimensions of self-similar measures and his inverse theorem for entropy. Our method can be seen as an extension of Hochman's theory from entropy to Lq norms, and likewise relies on an inverse theorem for the decay of Lq norms of discrete measures under convolution. This central piece of our approach may be of independent interest, and it is an application of well-known methods and results in additive combinatorics: the asymmetric version of the Balog-Szemerédi-Gowers Theorem due to Tao-Vu, and some constructions of Bourgain.
Fil: Shmerkin, Pablo Sebastian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina
Materia
BERNOULLI CONVOLUTIONS
DYNAMICAL RIGIDITY
INTERSECTIONS OF CANTOR SETS
SELF-SIMILAR MEASURES
×P -INVARIANT SETS
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:11336/128952

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spelling On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutionsShmerkin, Pablo SebastianBERNOULLI CONVOLUTIONSDYNAMICAL RIGIDITYINTERSECTIONS OF CANTOR SETSSELF-SIMILAR MEASURES×P -INVARIANT SETShttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the Lq dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg's long-standing conjecture on the dimension of the intersections of ×p- and ×q-invariant sets. Among several other applications, we also show that Bernoulli convolutions have an Lq density for all finite q, outside of a zero-dimensional set of exceptions. The proof of the main result is inspired by M. Hochman's approach to the dimensions of self-similar measures and his inverse theorem for entropy. Our method can be seen as an extension of Hochman's theory from entropy to Lq norms, and likewise relies on an inverse theorem for the decay of Lq norms of discrete measures under convolution. This central piece of our approach may be of independent interest, and it is an application of well-known methods and results in additive combinatorics: the asymmetric version of the Balog-Szemerédi-Gowers Theorem due to Tao-Vu, and some constructions of Bourgain.Fil: Shmerkin, Pablo Sebastian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; ArgentinaAnnal Mathematics2019-03info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/zipapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/128952Shmerkin, Pablo Sebastian; On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions; Annal Mathematics; Annals Of Mathematics; 189; 2; 3-2019; 319-3910003-486XCONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://annals.math.princeton.edu/2019/189-2/p01info:eu-repo/semantics/altIdentifier/doi/10.4007/annals.2019.189.2.1info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1609.07802info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T09:37:44Zoai:ri.conicet.gov.ar:11336/128952instacron: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 09:37:44.455CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions
title On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions
spellingShingle On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions
Shmerkin, Pablo Sebastian
BERNOULLI CONVOLUTIONS
DYNAMICAL RIGIDITY
INTERSECTIONS OF CANTOR SETS
SELF-SIMILAR MEASURES
×P -INVARIANT SETS
title_short On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions
title_full On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions
title_fullStr On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions
title_full_unstemmed On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions
title_sort On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of 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
DYNAMICAL RIGIDITY
INTERSECTIONS OF CANTOR SETS
SELF-SIMILAR MEASURES
×P -INVARIANT SETS
topic BERNOULLI CONVOLUTIONS
DYNAMICAL RIGIDITY
INTERSECTIONS OF CANTOR SETS
SELF-SIMILAR MEASURES
×P -INVARIANT SETS
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 study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the Lq dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg's long-standing conjecture on the dimension of the intersections of ×p- and ×q-invariant sets. Among several other applications, we also show that Bernoulli convolutions have an Lq density for all finite q, outside of a zero-dimensional set of exceptions. The proof of the main result is inspired by M. Hochman's approach to the dimensions of self-similar measures and his inverse theorem for entropy. Our method can be seen as an extension of Hochman's theory from entropy to Lq norms, and likewise relies on an inverse theorem for the decay of Lq norms of discrete measures under convolution. This central piece of our approach may be of independent interest, and it is an application of well-known methods and results in additive combinatorics: the asymmetric version of the Balog-Szemerédi-Gowers Theorem due to Tao-Vu, and some constructions of Bourgain.
Fil: Shmerkin, Pablo Sebastian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina
description We study a class of measures on the real line with a kind of self-similar structure, which we call dynamically driven self-similar measures, and contain proper self-similar measures such as Bernoulli convolutions as special cases. Our main result gives an expression for the Lq dimensions of such dynamically driven self-similar measures, under certain conditions. As an application, we settle Furstenberg's long-standing conjecture on the dimension of the intersections of ×p- and ×q-invariant sets. Among several other applications, we also show that Bernoulli convolutions have an Lq density for all finite q, outside of a zero-dimensional set of exceptions. The proof of the main result is inspired by M. Hochman's approach to the dimensions of self-similar measures and his inverse theorem for entropy. Our method can be seen as an extension of Hochman's theory from entropy to Lq norms, and likewise relies on an inverse theorem for the decay of Lq norms of discrete measures under convolution. This central piece of our approach may be of independent interest, and it is an application of well-known methods and results in additive combinatorics: the asymmetric version of the Balog-Szemerédi-Gowers Theorem due to Tao-Vu, and some constructions of Bourgain.
publishDate 2019
dc.date.none.fl_str_mv 2019-03
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/11336/128952
Shmerkin, Pablo Sebastian; On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions; Annal Mathematics; Annals Of Mathematics; 189; 2; 3-2019; 319-391
0003-486X
CONICET Digital
CONICET
url http://hdl.handle.net/11336/128952
identifier_str_mv Shmerkin, Pablo Sebastian; On Furstenberg's intersection conjecture, self-similar measures, and the Lq norms of convolutions; Annal Mathematics; Annals Of Mathematics; 189; 2; 3-2019; 319-391
0003-486X
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://annals.math.princeton.edu/2019/189-2/p01
info:eu-repo/semantics/altIdentifier/doi/10.4007/annals.2019.189.2.1
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1609.07802
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
application/pdf
dc.publisher.none.fl_str_mv Annal Mathematics
publisher.none.fl_str_mv Annal Mathematics
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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