Small Furstenberg sets

For α in (0, 1], a subset E of R2 is called Furstenberg set of type α or Fα-set if for each direction e in the unit circle there is a line segment `e in the direction of e such that the Hausdorff dimension of the set E ∩`e is greater than or equal to α. In this paper we use generalized Hausdorff mea...

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Detalles Bibliográficos
Autores: Molter, Úrsula María, Rela, Ezequiel
Tipo de recurso: artículo
Estado:Versión enviada para evaluación y publicación
Fecha de publicación:2013
País:España
Institución:Universidad de Sevilla (US)
Repositorio:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/47712
Acceso en línea:http://hdl.handle.net/11441/47712
https://doi.org/10.1016/j.jmaa.2012.11.001
Access Level:acceso abierto
Palabra clave:Furstenberg sets
Hausdorff dimension
Dimension function
Jarník’s theorems
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spelling Small Furstenberg setsMolter, Úrsula MaríaRela, EzequielFurstenberg setsHausdorff dimensionDimension functionJarník’s theoremsFor α in (0, 1], a subset E of R2 is called Furstenberg set of type α or Fα-set if for each direction e in the unit circle there is a line segment `e in the direction of e such that the Hausdorff dimension of the set E ∩`e 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 1/2. Since in a previous work we showed that 1/2 is a lower bound for the Hausdorff dimension of any E ∈ Fhγ, with the present construction, the value 1/2 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions hγ.Agencia Nacional de Promoción Científica y Tecnológica (Argentina)Universidad de Buenos AiresConsejo Nacional de Investigaciones Científicas y Técnicas (Argentina)ElsevierAnálisis Matemático2013info:eu-repo/semantics/articleinfo:eu-repo/semantics/submittedVersionapplication/pdfapplication/pdfhttp://hdl.handle.net/11441/47712https://doi.org/10.1016/j.jmaa.2012.11.001reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésJournal of Mathematical Analysis and Applications, 400 (2), 475-486.PICT2006-00177UBACyT X149PIP368http://ac.els-cdn.com/S0022247X12009055/1-s2.0-S0022247X12009055-main.pdf?_tid=b72d0526-952c-11e6-9697-00000aacb360&acdnat=1476793176_c50ef46000ead7f7c1d1e975f88f25f6info:eu-repo/semantics/openAccessoai:idus.us.es:11441/477122026-06-17T12:51:07Z
dc.title.none.fl_str_mv Small Furstenberg sets
title Small Furstenberg sets
spellingShingle Small Furstenberg sets
Molter, Úrsula María
Furstenberg sets
Hausdorff dimension
Dimension function
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, Úrsula María
Rela, Ezequiel
author Molter, Úrsula María
author_facet Molter, Úrsula María
Rela, Ezequiel
author_role author
author2 Rela, Ezequiel
author2_role author
dc.contributor.none.fl_str_mv Análisis Matemático
dc.subject.none.fl_str_mv Furstenberg sets
Hausdorff dimension
Dimension function
Jarník’s theorems
topic Furstenberg sets
Hausdorff dimension
Dimension function
Jarník’s theorems
description For α in (0, 1], a subset E of R2 is called Furstenberg set of type α or Fα-set if for each direction e in the unit circle there is a line segment `e in the direction of e such that the Hausdorff dimension of the set E ∩`e 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 1/2. Since in a previous work we showed that 1/2 is a lower bound for the Hausdorff dimension of any E ∈ Fhγ, with the present construction, the value 1/2 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions hγ.
publishDate 2013
dc.date.none.fl_str_mv 2013
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/submittedVersion
format article
status_str submittedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11441/47712
https://doi.org/10.1016/j.jmaa.2012.11.001
url http://hdl.handle.net/11441/47712
https://doi.org/10.1016/j.jmaa.2012.11.001
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Journal of Mathematical Analysis and Applications, 400 (2), 475-486.
PICT2006-00177
UBACyT X149
PIP368
http://ac.els-cdn.com/S0022247X12009055/1-s2.0-S0022247X12009055-main.pdf?_tid=b72d0526-952c-11e6-9697-00000aacb360&acdnat=1476793176_c50ef46000ead7f7c1d1e975f88f25f6
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:idUS. Depósito de Investigación de la Universidad de Sevilla
instname:Universidad de Sevilla (US)
instname_str Universidad de Sevilla (US)
reponame_str idUS. Depósito de Investigación de la Universidad de Sevilla
collection idUS. Depósito de Investigación de la Universidad de Sevilla
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15,300724

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