Furstenberg sets for a fractal set of directions

In this paper we study the behavior of the size of Furstenberg sets with respect to the size of the set of directions defining it. For any pair α, β ∈ (0, 1], we will say that a set E⊂ℝ 2 is an F αβ-set if there is a subset L of the unit circle of Hausdorff dimension at least β and, for each directi...

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Autores principales: Molter, Ursula Maria, Rela, Ezequiel
Publicado: 2012
Materias:
Dimension function
Furstenberg sets
Hausdorff dimension
Kakeya sets
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00029939_v140_n8_p2753_Molter
http://hdl.handle.net/20.500.12110/paper_00029939_v140_n8_p2753_Molter
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_00029939_v140_n8_p2753_Molter
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spelling paper:paper_00029939_v140_n8_p2753_Molter2023-06-08T14:23:32Z Furstenberg sets for a fractal set of directions Molter, Ursula Maria Rela, Ezequiel Dimension function Furstenberg sets Hausdorff dimension Kakeya sets In this paper we study the behavior of the size of Furstenberg sets with respect to the size of the set of directions defining it. For any pair α, β ∈ (0, 1], we will say that a set E⊂ℝ 2 is an F αβ-set if there is a subset L of the unit circle of Hausdorff dimension at least β and, for each direction e in L, there is a line segment ℓ e in the direction of e such that the Hausdorff dimension of the set E∩ℓ e is equal to or greater than α. The problem is considered in the wider scenario of generalized Hausdorff measures, giving estimates on the appropriate dimension functions for each class of Furstenberg sets. As a corollary of our main results, we obtain that dim(E) ≥ max {α + β/2; 2α + β-1} for any E ∈ F αβ. In particular we are able to extend previously known results to the "endpoint" α = 0 case. © 2011 American Mathematical Society. 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. 2012 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00029939_v140_n8_p2753_Molter http://hdl.handle.net/20.500.12110/paper_00029939_v140_n8_p2753_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
Kakeya sets
spellingShingle Dimension function
Furstenberg sets
Hausdorff dimension
Kakeya sets
Molter, Ursula Maria
Rela, Ezequiel
Furstenberg sets for a fractal set of directions
topic_facet Dimension function
Furstenberg sets
Hausdorff dimension
Kakeya sets
description In this paper we study the behavior of the size of Furstenberg sets with respect to the size of the set of directions defining it. For any pair α, β ∈ (0, 1], we will say that a set E⊂ℝ 2 is an F αβ-set if there is a subset L of the unit circle of Hausdorff dimension at least β and, for each direction e in L, there is a line segment ℓ e in the direction of e such that the Hausdorff dimension of the set E∩ℓ e is equal to or greater than α. The problem is considered in the wider scenario of generalized Hausdorff measures, giving estimates on the appropriate dimension functions for each class of Furstenberg sets. As a corollary of our main results, we obtain that dim(E) ≥ max {α + β/2; 2α + β-1} for any E ∈ F αβ. In particular we are able to extend previously known results to the "endpoint" α = 0 case. © 2011 American Mathematical Society.
author Molter, Ursula Maria
Rela, Ezequiel
author_facet Molter, Ursula Maria
Rela, Ezequiel
author_sort Molter, Ursula Maria
title Furstenberg sets for a fractal set of directions
title_short Furstenberg sets for a fractal set of directions
title_full Furstenberg sets for a fractal set of directions
title_fullStr Furstenberg sets for a fractal set of directions
title_full_unstemmed Furstenberg sets for a fractal set of directions
title_sort furstenberg sets for a fractal set of directions
publishDate 2012
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00029939_v140_n8_p2753_Molter
http://hdl.handle.net/20.500.12110/paper_00029939_v140_n8_p2753_Molter
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