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 w...

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Detalles Bibliográficos
Autores principales:	Molter, Ursula Maria, Rela, Ezequiel
Publicado:	2010
Materias:	Dimension function Furstenberg sets Hausdorff dimension
Acceso en línea:	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00018708_v223_n2_p672_Molter http://hdl.handle.net/20.500.12110/paper_00018708_v223_n2_p672_Molter
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	paper:paper_00018708_v223_n2_p672_Molter
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spelling	paper:paper_00018708_v223_n2_p672_Molter2023-06-08T14:21:45Z Improving dimension estimates for Furstenberg-type sets Molter, Ursula Maria Rela, Ezequiel Dimension function Furstenberg sets Hausdorff dimension 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. 2010 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00018708_v223_n2_p672_Molter http://hdl.handle.net/20.500.12110/paper_00018708_v223_n2_p672_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
spellingShingle	Dimension function Furstenberg sets Hausdorff dimension Molter, Ursula Maria Rela, Ezequiel Improving dimension estimates for Furstenberg-type sets
topic_facet	Dimension function Furstenberg sets Hausdorff dimension
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.
author	Molter, Ursula Maria Rela, Ezequiel
author_facet	Molter, Ursula Maria Rela, Ezequiel
author_sort	Molter, Ursula Maria
title	Improving dimension estimates for Furstenberg-type sets
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
publishDate	2010
url	https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00018708_v223_n2_p672_Molter http://hdl.handle.net/20.500.12110/paper_00018708_v223_n2_p672_Molter
work_keys_str_mv	AT molterursulamaria improvingdimensionestimatesforfurstenbergtypesets AT relaezequiel improvingdimensionestimatesforfurstenbergtypesets
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Improving dimension estimates for Furstenberg-type sets

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