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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todo:paper_00029939_v140_n8_p2753_Molter2023-10-03T13:55:13Z Furstenberg sets for a fractal set of directions Molter, U. Rela, E. 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. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar 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, U. Rela, E. 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. |
| format |
JOUR |
| author |
Molter, U. Rela, E. |
| author_facet |
Molter, U. Rela, E. |
| author_sort |
Molter, U. |
| 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 |
| url |
http://hdl.handle.net/20.500.12110/paper_00029939_v140_n8_p2753_Molter |
| work_keys_str_mv |
AT molteru furstenbergsetsforafractalsetofdirections AT relae furstenbergsetsforafractalsetofdirections |
| _version_ |
1807317323384619008 |