Patterns in random fractals

Autores
Shmerkin, Pablo Sebastian; Suomala, Ville
Año de publicación
2020
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We characterize the existence of certain geometric configurations in the fractal percolation limit set A in terms of the almost sure dimension of A. Some examples of the configurations we study are: homothetic copies of finite sets, angles, distances, and volumes of simplices. In the spirit of relative Szemer´edi theorems for random discrete sets, we also consider the corresponding problem for sets of positive ν-measure, where ν is the natural measure on A. In both cases we identify the dimension threshold for each class of configurations. These results are obtained by investigating the intersections of the products of m independent realizations of A with transversal planes and, more generally, algebraic varieties, and extend some well known features of independent percolation on trees to a setting with long-range dependencies.
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
PATTERNS
RANDOM FRACTALS
FRACTAL PERCOLATION
PROGRESSIONS
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:11336/168985
Acceder
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network_acronym_str CONICETDig
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network_name_str CONICET Digital (CONICET)
spelling Patterns in random fractalsShmerkin, Pablo SebastianSuomala, VillePATTERNSRANDOM FRACTALSFRACTAL PERCOLATIONPROGRESSIONShttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We characterize the existence of certain geometric configurations in the fractal percolation limit set A in terms of the almost sure dimension of A. Some examples of the configurations we study are: homothetic copies of finite sets, angles, distances, and volumes of simplices. In the spirit of relative Szemer´edi theorems for random discrete sets, we also consider the corresponding problem for sets of positive ν-measure, where ν is the natural measure on A. In both cases we identify the dimension threshold for each class of configurations. These results are obtained by investigating the intersections of the products of m independent realizations of A with transversal planes and, more generally, algebraic varieties, and extend some well known features of independent percolation on trees to a setting with long-range dependencies.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; FinlandiaJohns Hopkins University Press2020-06info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/168985Shmerkin, Pablo Sebastian; Suomala, Ville; Patterns in random fractals; Johns Hopkins University Press; American Journal Of Mathematics; 142; 3; 6-2020; 683-7491080-63770002-9327CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://muse.jhu.edu/article/755116info:eu-repo/semantics/altIdentifier/doi/10.1353/ajm.2020.0024info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1703.09553v1info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T09:57:07Zoai:ri.conicet.gov.ar:11336/168985instacron: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 09:57:07.366CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Patterns in random fractals
title Patterns in random fractals
spellingShingle Patterns in random fractals
Shmerkin, Pablo Sebastian
PATTERNS
RANDOM FRACTALS
FRACTAL PERCOLATION
PROGRESSIONS
title_short Patterns in random fractals
title_full Patterns in random fractals
title_fullStr Patterns in random fractals
title_full_unstemmed Patterns in random fractals
title_sort Patterns in random fractals
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 PATTERNS
RANDOM FRACTALS
FRACTAL PERCOLATION
PROGRESSIONS
topic PATTERNS
RANDOM FRACTALS
FRACTAL PERCOLATION
PROGRESSIONS
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 characterize the existence of certain geometric configurations in the fractal percolation limit set A in terms of the almost sure dimension of A. Some examples of the configurations we study are: homothetic copies of finite sets, angles, distances, and volumes of simplices. In the spirit of relative Szemer´edi theorems for random discrete sets, we also consider the corresponding problem for sets of positive ν-measure, where ν is the natural measure on A. In both cases we identify the dimension threshold for each class of configurations. These results are obtained by investigating the intersections of the products of m independent realizations of A with transversal planes and, more generally, algebraic varieties, and extend some well known features of independent percolation on trees to a setting with long-range dependencies.
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 characterize the existence of certain geometric configurations in the fractal percolation limit set A in terms of the almost sure dimension of A. Some examples of the configurations we study are: homothetic copies of finite sets, angles, distances, and volumes of simplices. In the spirit of relative Szemer´edi theorems for random discrete sets, we also consider the corresponding problem for sets of positive ν-measure, where ν is the natural measure on A. In both cases we identify the dimension threshold for each class of configurations. These results are obtained by investigating the intersections of the products of m independent realizations of A with transversal planes and, more generally, algebraic varieties, and extend some well known features of independent percolation on trees to a setting with long-range dependencies.
publishDate 2020
dc.date.none.fl_str_mv 2020-06
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/11336/168985
Shmerkin, Pablo Sebastian; Suomala, Ville; Patterns in random fractals; Johns Hopkins University Press; American Journal Of Mathematics; 142; 3; 6-2020; 683-749
1080-6377
0002-9327
CONICET Digital
CONICET
url http://hdl.handle.net/11336/168985
identifier_str_mv Shmerkin, Pablo Sebastian; Suomala, Ville; Patterns in random fractals; Johns Hopkins University Press; American Journal Of Mathematics; 142; 3; 6-2020; 683-749
1080-6377
0002-9327
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://muse.jhu.edu/article/755116
info:eu-repo/semantics/altIdentifier/doi/10.1353/ajm.2020.0024
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1703.09553v1
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/pdf
dc.publisher.none.fl_str_mv Johns Hopkins University Press
publisher.none.fl_str_mv Johns Hopkins University Press
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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