Furstenberg sets for a fractal set of directions

Autores
Molter, Ursula Maria; Rela, Ezequiel
Año de publicación
2012
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
In this note we study the behavior of the size of Furstenberg sets with respect to the size of the set of directions defining it. For any pair $alpha,etain(0,1]$, we will say that a set $Esubset R^2$ is an $F_{alphaeta}$-set if there is a subset $L$ of the unit circle of Hausdorff dimension at least $eta$ and, for each direction $e$ in $L$, there is a line segment $ell_e$ in the direction of $e$ such that the Hausdorff dimension of the set $Ecapell_e$ is equal or greater than $alpha$. The problem is considered in the wider scenario of generalized Hausdorff measures, giving estimates on the appropriate dimension functions for each class of Furstenberg sets. As a corollary of our main results, we obtain that $dim(E)gemaxleft{alpha+rac{eta}{2} ; 2alpha+eta -1 ight}$ for any $Ein F_{alphaeta}$. In particular we are able to extend previously known results to the ``endpoint´´ $alpha=0$ case.
Fil: Molter, Ursula Maria. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
Fil: Rela, Ezequiel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
Materia
HAUSDORFF DIMENSION
FURSTENBERG SET
KAKEYA SET
DIMENSION FUNCTION
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:11336/265643
Acceder
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network_name_str CONICET Digital (CONICET)
spelling Furstenberg sets for a fractal set of directionsMolter, Ursula MariaRela, EzequielHAUSDORFF DIMENSIONFURSTENBERG SETKAKEYA SETDIMENSION FUNCTIONhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1In this note we study the behavior of the size of Furstenberg sets with respect to the size of the set of directions defining it. For any pair $alpha,etain(0,1]$, we will say that a set $Esubset R^2$ is an $F_{alphaeta}$-set if there is a subset $L$ of the unit circle of Hausdorff dimension at least $eta$ and, for each direction $e$ in $L$, there is a line segment $ell_e$ in the direction of $e$ such that the Hausdorff dimension of the set $Ecapell_e$ is equal or greater than $alpha$. The problem is considered in the wider scenario of generalized Hausdorff measures, giving estimates on the appropriate dimension functions for each class of Furstenberg sets. As a corollary of our main results, we obtain that $dim(E)gemaxleft{alpha+rac{eta}{2} ; 2alpha+eta -1 ight}$ for any $Ein F_{alphaeta}$. In particular we are able to extend previously known results to the ``endpoint´´ $alpha=0$ case.Fil: Molter, Ursula Maria. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Rela, Ezequiel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaAmerican Mathematical Society2012-12info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/265643Molter, Ursula Maria; Rela, Ezequiel; Furstenberg sets for a fractal set of directions; American Mathematical Society; Proceedings of the American Mathematical Society; 140; 8; 12-2012; 2753-27650002-9939CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/http://www.ams.org/journals/proc/0000-000-00/S0002-9939-2011-11111-0/info:eu-repo/semantics/altIdentifier/doi/10.1090/S0002-9939-2011-11111-0info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-10-15T14:35:54Zoai:ri.conicet.gov.ar:11336/265643instacron: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:34982025-10-15 14:35:54.347CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Furstenberg sets for a fractal set of directions
title Furstenberg sets for a fractal set of directions
spellingShingle Furstenberg sets for a fractal set of directions
Molter, Ursula Maria
HAUSDORFF DIMENSION
FURSTENBERG SET
KAKEYA SET
DIMENSION FUNCTION
title_short Furstenberg sets for a fractal set of directions
title_full Furstenberg sets for a fractal set of directions
title_fullStr Furstenberg sets for a fractal set of directions
title_full_unstemmed Furstenberg sets for a fractal set of directions
title_sort Furstenberg sets for a fractal set of directions
dc.creator.none.fl_str_mv Molter, Ursula Maria
Rela, Ezequiel
author Molter, Ursula Maria
author_facet Molter, Ursula Maria
Rela, Ezequiel
author_role author
author2 Rela, Ezequiel
author2_role author
dc.subject.none.fl_str_mv HAUSDORFF DIMENSION
FURSTENBERG SET
KAKEYA SET
DIMENSION FUNCTION
topic HAUSDORFF DIMENSION
FURSTENBERG SET
KAKEYA SET
DIMENSION FUNCTION
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv In this note we study the behavior of the size of Furstenberg sets with respect to the size of the set of directions defining it. For any pair $alpha,etain(0,1]$, we will say that a set $Esubset R^2$ is an $F_{alphaeta}$-set if there is a subset $L$ of the unit circle of Hausdorff dimension at least $eta$ and, for each direction $e$ in $L$, there is a line segment $ell_e$ in the direction of $e$ such that the Hausdorff dimension of the set $Ecapell_e$ is equal or greater than $alpha$. The problem is considered in the wider scenario of generalized Hausdorff measures, giving estimates on the appropriate dimension functions for each class of Furstenberg sets. As a corollary of our main results, we obtain that $dim(E)gemaxleft{alpha+rac{eta}{2} ; 2alpha+eta -1 ight}$ for any $Ein F_{alphaeta}$. In particular we are able to extend previously known results to the ``endpoint´´ $alpha=0$ case.
Fil: Molter, Ursula Maria. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
Fil: Rela, Ezequiel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina
description In this note we study the behavior of the size of Furstenberg sets with respect to the size of the set of directions defining it. For any pair $alpha,etain(0,1]$, we will say that a set $Esubset R^2$ is an $F_{alphaeta}$-set if there is a subset $L$ of the unit circle of Hausdorff dimension at least $eta$ and, for each direction $e$ in $L$, there is a line segment $ell_e$ in the direction of $e$ such that the Hausdorff dimension of the set $Ecapell_e$ is equal or greater than $alpha$. The problem is considered in the wider scenario of generalized Hausdorff measures, giving estimates on the appropriate dimension functions for each class of Furstenberg sets. As a corollary of our main results, we obtain that $dim(E)gemaxleft{alpha+rac{eta}{2} ; 2alpha+eta -1 ight}$ for any $Ein F_{alphaeta}$. In particular we are able to extend previously known results to the ``endpoint´´ $alpha=0$ case.
publishDate 2012
dc.date.none.fl_str_mv 2012-12
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/11336/265643
Molter, Ursula Maria; Rela, Ezequiel; Furstenberg sets for a fractal set of directions; American Mathematical Society; Proceedings of the American Mathematical Society; 140; 8; 12-2012; 2753-2765
0002-9939
CONICET Digital
CONICET
url http://hdl.handle.net/11336/265643
identifier_str_mv Molter, Ursula Maria; Rela, Ezequiel; Furstenberg sets for a fractal set of directions; American Mathematical Society; Proceedings of the American Mathematical Society; 140; 8; 12-2012; 2753-2765
0002-9939
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/journals/proc/0000-000-00/S0002-9939-2011-11111-0/
info:eu-repo/semantics/altIdentifier/doi/10.1090/S0002-9939-2011-11111-0
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
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 13.22299

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