Small Furstenberg sets

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 me...

Descripción completa

Guardado en:
Detalles Bibliográficos
Autores principales: Molter, Ursula Maria, Rela, Ezequiel
Publicado: 2013
Materias:
Dimension function
Furstenberg sets
Hausdorff dimension
Jarník's theorems
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0022247X_v400_n2_p475_Molter
http://hdl.handle.net/20.500.12110/paper_0022247X_v400_n2_p475_Molter
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_0022247X_v400_n2_p475_Molter
record_format dspace
spelling paper:paper_0022247X_v400_n2_p475_Molter2025-07-30T17:30:04Z Small Furstenberg sets Molter, Ursula Maria Rela, Ezequiel Dimension function Furstenberg sets Hausdorff dimension Jarník's theorems 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. 2013 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0022247X_v400_n2_p475_Molter http://hdl.handle.net/20.500.12110/paper_0022247X_v400_n2_p475_Molter
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Dimension function
Furstenberg sets
Hausdorff dimension
Jarník's theorems
spellingShingle Dimension function
Furstenberg sets
Hausdorff dimension
Jarník's theorems
Molter, Ursula Maria
Rela, Ezequiel
Small Furstenberg sets
topic_facet Dimension function
Furstenberg sets
Hausdorff dimension
Jarník's theorems
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.
author Molter, Ursula Maria
Rela, Ezequiel
author_facet Molter, Ursula Maria
Rela, Ezequiel
author_sort Molter, Ursula Maria
title Small Furstenberg sets
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
publishDate 2013
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0022247X_v400_n2_p475_Molter
http://hdl.handle.net/20.500.12110/paper_0022247X_v400_n2_p475_Molter
work_keys_str_mv AT molterursulamaria smallfurstenbergsets
AT relaezequiel smallfurstenbergsets
_version_ 1840325542782435328

Ejemplares similares