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Furstenberg sets for a fractal set of directions

 

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Author: Molter, Úrsula María
Rela, Ezequiel
Department: Universidad de Sevilla. Departamento de Análisis Matemático
Date: 2012-08
Published in: Proceedings of the American Mathematical Society, 140 (8), 2753-2765.
Document type: Article
Abstract: 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 ⊂ R2 is an Fαβ-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 e in the direction of e such that the Hausdorff dimension of the set E∩ 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 ∈ Fαβ. In particular we are able to extend previously known results to the “endpoint” α = 0 case.
Cite: Molter, Ú.M. y Rela, E. (2012). Furstenberg sets for a fractal set of directions. Proceedings of the American Mathematical Society, 140 (8), 2753-2765.
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URI: http://hdl.handle.net/11441/47719

DOI: 10.1090/S0002-9939-2011-11111-0

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