Improving dimension estimates for Furstenberg-type sets
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 know...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión enviada para evaluación y publicación |
| Fecha de publicación: | 2010 |
| País: | España |
| Institución: | Universidad de Sevilla (US) |
| Repositorio: | idUS. Depósito de Investigación de la Universidad de Sevilla |
| OAI Identifier: | oai:idus.us.es:11441/47720 |
| Acceso en línea: | http://hdl.handle.net/11441/47720 https://doi.org/10.1016/j.aim.2009.08.019 |
| Access Level: | acceso abierto |
| Palabra clave: | Furstenberg sets Hausdorff dimension Dimension function |
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Improving dimension estimates for Furstenberg-type setsMolter, Úrsula MaríaRela, EzequielFurstenberg setsHausdorff dimensionDimension functionIn 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 , 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.Agencia Nacional de Promoción Científica y Tecnológica (Argentina)Universidad de Buenos AiresElsevierAnálisis MatemáticoAgencia Nacional de Promoción Científica y Tecnológica. ArgentinaUniversidad de Buenos Aires2010info:eu-repo/semantics/articleinfo:eu-repo/semantics/submittedVersionapplication/pdfapplication/pdfhttp://hdl.handle.net/11441/47720https://doi.org/10.1016/j.aim.2009.08.019reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésAdvances in Mathematics, 223 (2), 672-688.PICT2006-00177UBACyT X149http://ac.els-cdn.com/S0001870809002667/1-s2.0-S0001870809002667-main.pdf?_tid=4445e7d4-95c3-11e6-aa03-00000aab0f26&acdnat=1476857837_1cb593375dcb252175720f3c881b1704info:eu-repo/semantics/openAccessoai:idus.us.es:11441/477202026-06-17T12:51:07Z |
| 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, Úrsula María Furstenberg sets Hausdorff dimension Dimension function |
| 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, 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 Agencia Nacional de Promoción Científica y Tecnológica. Argentina Universidad de Buenos Aires |
| dc.subject.none.fl_str_mv |
Furstenberg sets Hausdorff dimension Dimension function |
| topic |
Furstenberg sets Hausdorff dimension Dimension function |
| 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 , 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. |
| publishDate |
2010 |
| dc.date.none.fl_str_mv |
2010 |
| dc.type.none.fl_str_mv |
info:eu-repo/semantics/article info:eu-repo/semantics/submittedVersion |
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article |
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submittedVersion |
| dc.identifier.none.fl_str_mv |
http://hdl.handle.net/11441/47720 https://doi.org/10.1016/j.aim.2009.08.019 |
| url |
http://hdl.handle.net/11441/47720 https://doi.org/10.1016/j.aim.2009.08.019 |
| dc.language.none.fl_str_mv |
Inglés |
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Inglés |
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Advances in Mathematics, 223 (2), 672-688. PICT2006-00177 UBACyT X149 http://ac.els-cdn.com/S0001870809002667/1-s2.0-S0001870809002667-main.pdf?_tid=4445e7d4-95c3-11e6-aa03-00000aab0f26&acdnat=1476857837_1cb593375dcb252175720f3c881b1704 |
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info:eu-repo/semantics/openAccess |
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openAccess |
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application/pdf application/pdf |
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Elsevier |
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Elsevier |
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reponame:idUS. Depósito de Investigación de la Universidad de Sevilla instname:Universidad de Sevilla (US) |
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