Improving dimension estimates for Furstenberg-type sets

Autores
Molter, U.; Rela, E.
Año de publicación
2010
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
In this paper we study the problem of estimating the generalized Hausdorff dimension of Furstenberg sets in the plane. For α ∈ (0, 1], a set F in the plane is said to be an α-Furstenberg set if for each direction e there is a line segment ℓe in the direction of e for which dimH (ℓe ∩ F) ≥ α. It is well known that dimH (F) ≥ max {2 α, α + frac(1, 2)}, and it is also known that these sets can have zero measure at their critical dimension. By looking at general Hausdorff measures Hh defined for doubling functions, that need not be power laws, we obtain finer estimates for the size of the more general h-Furstenberg sets. Further, this approach allow us to sharpen the known bounds on the dimension of classical Furstenberg sets. The main difficulty we had to overcome, was that if Hh (F) = 0, there always exists g ≺ h such that Hg (F) = 0 (here ≺ refers to the natural ordering on general Hausdorff dimension functions). Hence, in order to estimate the measure of general Furstenberg sets, we have to consider dimension functions that are a true step down from the critical one. We provide rather precise estimates on the size of this step and by doing so, we can include a family of zero dimensional Furstenberg sets associated to dimension functions that grow faster than any power function at zero. With some additional growth conditions on these zero dimensional functions, we extend the known inequalities to include the endpoint α = 0. © 2009 Elsevier Inc. All rights reserved.
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
Adv. Math. 2010;223(2):672-688
Materia
Dimension function
Furstenberg sets
Hausdorff dimension
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_00018708_v223_n2_p672_Molter

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network_name_str Biblioteca Digital (UBA-FCEN)
spelling Improving dimension estimates for Furstenberg-type setsMolter, U.Rela, E.Dimension functionFurstenberg setsHausdorff dimensionIn this paper we study the problem of estimating the generalized Hausdorff dimension of Furstenberg sets in the plane. For α ∈ (0, 1], a set F in the plane is said to be an α-Furstenberg set if for each direction e there is a line segment ℓe in the direction of e for which dimH (ℓe ∩ F) ≥ α. It is well known that dimH (F) ≥ max {2 α, α + frac(1, 2)}, and it is also known that these sets can have zero measure at their critical dimension. By looking at general Hausdorff measures Hh defined for doubling functions, that need not be power laws, we obtain finer estimates for the size of the more general h-Furstenberg sets. Further, this approach allow us to sharpen the known bounds on the dimension of classical Furstenberg sets. The main difficulty we had to overcome, was that if Hh (F) = 0, there always exists g ≺ h such that Hg (F) = 0 (here ≺ refers to the natural ordering on general Hausdorff dimension functions). Hence, in order to estimate the measure of general Furstenberg sets, we have to consider dimension functions that are a true step down from the critical one. We provide rather precise estimates on the size of this step and by doing so, we can include a family of zero dimensional Furstenberg sets associated to dimension functions that grow faster than any power function at zero. With some additional growth conditions on these zero dimensional functions, we extend the known inequalities to include the endpoint α = 0. © 2009 Elsevier Inc. All rights reserved.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.2010info: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_00018708_v223_n2_p672_MolterAdv. Math. 2010;223(2):672-688reponame: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/ar2025-09-29T13:42:50Zpaperaa:paper_00018708_v223_n2_p672_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:18962025-09-29 13:42:50.86Biblioteca Digital (UBA-FCEN) - Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturalesfalse
dc.title.none.fl_str_mv Improving dimension estimates for Furstenberg-type sets
title Improving dimension estimates for Furstenberg-type sets
spellingShingle Improving dimension estimates for Furstenberg-type sets
Molter, U.
Dimension function
Furstenberg sets
Hausdorff dimension
title_short Improving dimension estimates for Furstenberg-type sets
title_full Improving dimension estimates for Furstenberg-type sets
title_fullStr Improving dimension estimates for Furstenberg-type sets
title_full_unstemmed Improving dimension estimates for Furstenberg-type sets
title_sort Improving dimension estimates for Furstenberg-type 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
topic Dimension function
Furstenberg sets
Hausdorff dimension
dc.description.none.fl_txt_mv In this paper we study the problem of estimating the generalized Hausdorff dimension of Furstenberg sets in the plane. For α ∈ (0, 1], a set F in the plane is said to be an α-Furstenberg set if for each direction e there is a line segment ℓe in the direction of e for which dimH (ℓe ∩ F) ≥ α. It is well known that dimH (F) ≥ max {2 α, α + frac(1, 2)}, and it is also known that these sets can have zero measure at their critical dimension. By looking at general Hausdorff measures Hh defined for doubling functions, that need not be power laws, we obtain finer estimates for the size of the more general h-Furstenberg sets. Further, this approach allow us to sharpen the known bounds on the dimension of classical Furstenberg sets. The main difficulty we had to overcome, was that if Hh (F) = 0, there always exists g ≺ h such that Hg (F) = 0 (here ≺ refers to the natural ordering on general Hausdorff dimension functions). Hence, in order to estimate the measure of general Furstenberg sets, we have to consider dimension functions that are a true step down from the critical one. We provide rather precise estimates on the size of this step and by doing so, we can include a family of zero dimensional Furstenberg sets associated to dimension functions that grow faster than any power function at zero. With some additional growth conditions on these zero dimensional functions, we extend the known inequalities to include the endpoint α = 0. © 2009 Elsevier Inc. All rights reserved.
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 In this paper we study the problem of estimating the generalized Hausdorff dimension of Furstenberg sets in the plane. For α ∈ (0, 1], a set F in the plane is said to be an α-Furstenberg set if for each direction e there is a line segment ℓe in the direction of e for which dimH (ℓe ∩ F) ≥ α. It is well known that dimH (F) ≥ max {2 α, α + frac(1, 2)}, and it is also known that these sets can have zero measure at their critical dimension. By looking at general Hausdorff measures Hh defined for doubling functions, that need not be power laws, we obtain finer estimates for the size of the more general h-Furstenberg sets. Further, this approach allow us to sharpen the known bounds on the dimension of classical Furstenberg sets. The main difficulty we had to overcome, was that if Hh (F) = 0, there always exists g ≺ h such that Hg (F) = 0 (here ≺ refers to the natural ordering on general Hausdorff dimension functions). Hence, in order to estimate the measure of general Furstenberg sets, we have to consider dimension functions that are a true step down from the critical one. We provide rather precise estimates on the size of this step and by doing so, we can include a family of zero dimensional Furstenberg sets associated to dimension functions that grow faster than any power function at zero. With some additional growth conditions on these zero dimensional functions, we extend the known inequalities to include the endpoint α = 0. © 2009 Elsevier Inc. All rights reserved.
publishDate 2010
dc.date.none.fl_str_mv 2010
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
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info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/20.500.12110/paper_00018708_v223_n2_p672_Molter
url http://hdl.handle.net/20.500.12110/paper_00018708_v223_n2_p672_Molter
dc.language.none.fl_str_mv eng
language eng
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
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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 Adv. Math. 2010;223(2):672-688
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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