Small Furstenberg sets

Autores
Molter, U.; Rela, E.
Año de publicación
2013
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
For α in (0, 1], a subset E of R2 is called a Furstenberg set of type α or Fα-set if for each direction e in the unit circle there is a line segment ℓe in the direction of e such that the Hausdorff dimension of the set E∩ℓe is greater than or equal to α. In this paper we use generalized Hausdorff measures to give estimates on the size of these sets. Our main result is to obtain a sharp dimension estimate for a whole class of zero-dimensional Furstenberg type sets. Namely, for hγ(x)=log-γ(1x), γ>0, we construct a set Eγ∈Fhγ of Hausdorff dimension not greater than 12. Since in a previous work we showed that 12 is a lower bound for the Hausdorff dimension of any E∈Fhγ, with the present construction, the value 12 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functionshγ. © 2012 Elsevier Ltd.
Fil:Molter, U. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.
Fil:Rela, E. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.
Fuente
J. Math. Anal. Appl. 2013;400(2):475-486
Materia
Dimension function
Furstenberg sets
Hausdorff dimension
Jarník's theorems
Nivel de accesibilidad
acceso abierto
Condiciones de uso
http://creativecommons.org/licenses/by/2.5/ar
Repositorio
Biblioteca Digital (UBA-FCEN)
Institución
Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales
OAI Identificador
paperaa:paper_0022247X_v400_n2_p475_Molter
Acceder
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oai_identifier_str paperaa:paper_0022247X_v400_n2_p475_Molter
network_acronym_str BDUBAFCEN
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network_name_str Biblioteca Digital (UBA-FCEN)
spelling Small Furstenberg setsMolter, U.Rela, E.Dimension functionFurstenberg setsHausdorff dimensionJarník's theoremsFor α in (0, 1], a subset E of R2 is called a Furstenberg set of type α or Fα-set if for each direction e in the unit circle there is a line segment ℓe in the direction of e such that the Hausdorff dimension of the set E∩ℓe is greater than or equal to α. In this paper we use generalized Hausdorff measures to give estimates on the size of these sets. Our main result is to obtain a sharp dimension estimate for a whole class of zero-dimensional Furstenberg type sets. Namely, for hγ(x)=log-γ(1x), γ>0, we construct a set Eγ∈Fhγ of Hausdorff dimension not greater than 12. Since in a previous work we showed that 12 is a lower bound for the Hausdorff dimension of any E∈Fhγ, with the present construction, the value 12 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functionshγ. © 2012 Elsevier Ltd.Fil:Molter, U. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.Fil:Rela, E. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.2013info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfhttp://hdl.handle.net/20.500.12110/paper_0022247X_v400_n2_p475_MolterJ. Math. Anal. Appl. 2013;400(2):475-486reponame:Biblioteca Digital (UBA-FCEN)instname:Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturalesinstacron:UBA-FCENenginfo:eu-repo/semantics/openAccesshttp://creativecommons.org/licenses/by/2.5/ar2025-10-16T09:30:01Zpaperaa:paper_0022247X_v400_n2_p475_MolterInstitucionalhttps://digital.bl.fcen.uba.ar/Universidad públicaNo correspondehttps://digital.bl.fcen.uba.ar/cgi-bin/oaiserver.cgiana@bl.fcen.uba.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:18962025-10-16 09:30:02.97Biblioteca Digital (UBA-FCEN) - Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturalesfalse
dc.title.none.fl_str_mv Small Furstenberg sets
title Small Furstenberg sets
spellingShingle Small Furstenberg sets
Molter, U.
Dimension function
Furstenberg sets
Hausdorff dimension
Jarník's theorems
title_short Small Furstenberg sets
title_full Small Furstenberg sets
title_fullStr Small Furstenberg sets
title_full_unstemmed Small Furstenberg sets
title_sort Small Furstenberg sets
dc.creator.none.fl_str_mv Molter, U.
Rela, E.
author Molter, U.
author_facet Molter, U.
Rela, E.
author_role author
author2 Rela, E.
author2_role author
dc.subject.none.fl_str_mv Dimension function
Furstenberg sets
Hausdorff dimension
Jarník's theorems
topic Dimension function
Furstenberg sets
Hausdorff dimension
Jarník's theorems
dc.description.none.fl_txt_mv For α in (0, 1], a subset E of R2 is called a Furstenberg set of type α or Fα-set if for each direction e in the unit circle there is a line segment ℓe in the direction of e such that the Hausdorff dimension of the set E∩ℓe is greater than or equal to α. In this paper we use generalized Hausdorff measures to give estimates on the size of these sets. Our main result is to obtain a sharp dimension estimate for a whole class of zero-dimensional Furstenberg type sets. Namely, for hγ(x)=log-γ(1x), γ>0, we construct a set Eγ∈Fhγ of Hausdorff dimension not greater than 12. Since in a previous work we showed that 12 is a lower bound for the Hausdorff dimension of any E∈Fhγ, with the present construction, the value 12 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functionshγ. © 2012 Elsevier Ltd.
Fil:Molter, U. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.
Fil:Rela, E. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina.
description For α in (0, 1], a subset E of R2 is called a Furstenberg set of type α or Fα-set if for each direction e in the unit circle there is a line segment ℓe in the direction of e such that the Hausdorff dimension of the set E∩ℓe is greater than or equal to α. In this paper we use generalized Hausdorff measures to give estimates on the size of these sets. Our main result is to obtain a sharp dimension estimate for a whole class of zero-dimensional Furstenberg type sets. Namely, for hγ(x)=log-γ(1x), γ>0, we construct a set Eγ∈Fhγ of Hausdorff dimension not greater than 12. Since in a previous work we showed that 12 is a lower bound for the Hausdorff dimension of any E∈Fhγ, with the present construction, the value 12 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functionshγ. © 2012 Elsevier Ltd.
publishDate 2013
dc.date.none.fl_str_mv 2013
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/20.500.12110/paper_0022247X_v400_n2_p475_Molter
url http://hdl.handle.net/20.500.12110/paper_0022247X_v400_n2_p475_Molter
dc.language.none.fl_str_mv eng
language eng
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by/2.5/ar
eu_rights_str_mv openAccess
rights_invalid_str_mv http://creativecommons.org/licenses/by/2.5/ar
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv J. Math. Anal. Appl. 2013;400(2):475-486
reponame:Biblioteca Digital (UBA-FCEN)
instname:Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales
instacron:UBA-FCEN
reponame_str Biblioteca Digital (UBA-FCEN)
collection Biblioteca Digital (UBA-FCEN)
instname_str Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales
instacron_str UBA-FCEN
institution UBA-FCEN
repository.name.fl_str_mv Biblioteca Digital (UBA-FCEN) - Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales
repository.mail.fl_str_mv ana@bl.fcen.uba.ar
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