Spatially independent martingales, intersections and applications

Autores
Shmerkin, Pablo Sebastian; Suomala, Ville
Año de publicación
2018
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We define a class of random measures, spatially independent martingales, which we view as a natural generalization of the canonical random discrete set, and which includes as special cases many variants of fractal percolation and Poissonian cut-outs. We pair the random measures with deterministic families of parametrized measures {ηt}t, and show that under some natural checkable conditions, a.s. the mass of the intersections is H¨older continuous as a function of t. This continuity phenomenon turns out to underpin a large amount of geometric information about these measures, allowing us to unify and substantially generalize a large number of existing results on the geometry of random Cantor sets and measures, as well as obtaining many new ones. Among other things, for large classes of random fractals we establish (a) very strong versions of the Marstrand-Mattila projection and slicing results, as well as dimension conservation, (b) slicing results with respect to algebraic curves and self-similar sets, (c) smoothness of convolutions of measures, including self-convolutions, and nonempty interior for sumsets, (d) rapid Fourier decay. Among other applications, we obtain an answer to a question of I. Laba in connection to the restriction problem for fractal measures.
Fil: Shmerkin, Pablo Sebastian. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Suomala, Ville. Universidad de Oulu; Finlandia
Materia
MARTINGALES
RANDOM MEASURES
FRACTAL PERCOLATION
HAUSDORFF DIMENSION
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:11336/99802
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/99802
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling Spatially independent martingales, intersections and applicationsShmerkin, Pablo SebastianSuomala, VilleMARTINGALESRANDOM MEASURESFRACTAL PERCOLATIONHAUSDORFF DIMENSIONhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We define a class of random measures, spatially independent martingales, which we view as a natural generalization of the canonical random discrete set, and which includes as special cases many variants of fractal percolation and Poissonian cut-outs. We pair the random measures with deterministic families of parametrized measures {ηt}t, and show that under some natural checkable conditions, a.s. the mass of the intersections is H¨older continuous as a function of t. This continuity phenomenon turns out to underpin a large amount of geometric information about these measures, allowing us to unify and substantially generalize a large number of existing results on the geometry of random Cantor sets and measures, as well as obtaining many new ones. Among other things, for large classes of random fractals we establish (a) very strong versions of the Marstrand-Mattila projection and slicing results, as well as dimension conservation, (b) slicing results with respect to algebraic curves and self-similar sets, (c) smoothness of convolutions of measures, including self-convolutions, and nonempty interior for sumsets, (d) rapid Fourier decay. Among other applications, we obtain an answer to a question of I. Laba in connection to the restriction problem for fractal measures.Fil: Shmerkin, Pablo Sebastian. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Suomala, Ville. Universidad de Oulu; FinlandiaAmerican Mathematical Society2018-01info: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/11336/99802Shmerkin, Pablo Sebastian; Suomala, Ville; Spatially independent martingales, intersections and applications; American Mathematical Society; Memoirs Of The American Mathematical Society (ams); 251; 1195; 1-2018; 1-960065-9266CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/http://www.ams.org/books/memo/1195/info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1409.6707v4info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:24:41Zoai:ri.conicet.gov.ar:11336/99802instacron: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 10:24:41.312CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Spatially independent martingales, intersections and applications
title Spatially independent martingales, intersections and applications
spellingShingle Spatially independent martingales, intersections and applications
Shmerkin, Pablo Sebastian
MARTINGALES
RANDOM MEASURES
FRACTAL PERCOLATION
HAUSDORFF DIMENSION
title_short Spatially independent martingales, intersections and applications
title_full Spatially independent martingales, intersections and applications
title_fullStr Spatially independent martingales, intersections and applications
title_full_unstemmed Spatially independent martingales, intersections and applications
title_sort Spatially independent martingales, intersections and applications
dc.creator.none.fl_str_mv Shmerkin, Pablo Sebastian
Suomala, Ville
author Shmerkin, Pablo Sebastian
author_facet Shmerkin, Pablo Sebastian
Suomala, Ville
author_role author
author2 Suomala, Ville
author2_role author
dc.subject.none.fl_str_mv MARTINGALES
RANDOM MEASURES
FRACTAL PERCOLATION
HAUSDORFF DIMENSION
topic MARTINGALES
RANDOM MEASURES
FRACTAL PERCOLATION
HAUSDORFF DIMENSION
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 define a class of random measures, spatially independent martingales, which we view as a natural generalization of the canonical random discrete set, and which includes as special cases many variants of fractal percolation and Poissonian cut-outs. We pair the random measures with deterministic families of parametrized measures {ηt}t, and show that under some natural checkable conditions, a.s. the mass of the intersections is H¨older continuous as a function of t. This continuity phenomenon turns out to underpin a large amount of geometric information about these measures, allowing us to unify and substantially generalize a large number of existing results on the geometry of random Cantor sets and measures, as well as obtaining many new ones. Among other things, for large classes of random fractals we establish (a) very strong versions of the Marstrand-Mattila projection and slicing results, as well as dimension conservation, (b) slicing results with respect to algebraic curves and self-similar sets, (c) smoothness of convolutions of measures, including self-convolutions, and nonempty interior for sumsets, (d) rapid Fourier decay. Among other applications, we obtain an answer to a question of I. Laba in connection to the restriction problem for fractal measures.
Fil: Shmerkin, Pablo Sebastian. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Fil: Suomala, Ville. Universidad de Oulu; Finlandia
description We define a class of random measures, spatially independent martingales, which we view as a natural generalization of the canonical random discrete set, and which includes as special cases many variants of fractal percolation and Poissonian cut-outs. We pair the random measures with deterministic families of parametrized measures {ηt}t, and show that under some natural checkable conditions, a.s. the mass of the intersections is H¨older continuous as a function of t. This continuity phenomenon turns out to underpin a large amount of geometric information about these measures, allowing us to unify and substantially generalize a large number of existing results on the geometry of random Cantor sets and measures, as well as obtaining many new ones. Among other things, for large classes of random fractals we establish (a) very strong versions of the Marstrand-Mattila projection and slicing results, as well as dimension conservation, (b) slicing results with respect to algebraic curves and self-similar sets, (c) smoothness of convolutions of measures, including self-convolutions, and nonempty interior for sumsets, (d) rapid Fourier decay. Among other applications, we obtain an answer to a question of I. Laba in connection to the restriction problem for fractal measures.
publishDate 2018
dc.date.none.fl_str_mv 2018-01
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/11336/99802
Shmerkin, Pablo Sebastian; Suomala, Ville; Spatially independent martingales, intersections and applications; American Mathematical Society; Memoirs Of The American Mathematical Society (ams); 251; 1195; 1-2018; 1-96
0065-9266
CONICET Digital
CONICET
url http://hdl.handle.net/11336/99802
identifier_str_mv Shmerkin, Pablo Sebastian; Suomala, Ville; Spatially independent martingales, intersections and applications; American Mathematical Society; Memoirs Of The American Mathematical Society (ams); 251; 1195; 1-2018; 1-96
0065-9266
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://www.ams.org/books/memo/1195/
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1409.6707v4
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 American Mathematical Society
publisher.none.fl_str_mv American Mathematical Society
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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