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
- Institución
- Consejo Nacional de Investigaciones Científicas y Técnicas
- OAI Identificador
- oai:ri.conicet.gov.ar:11336/128952
Ver los metadatos del registro completo
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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) |
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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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