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Refined size estimates for Furstenberg sets via Hausdorff measures: a survey of some recent results

Opened Access Refined size estimates for Furstenberg sets via Hausdorff measures: a survey of some recent results

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Autor: Rela, Ezequiel
Coordinador/Director: Cepedello Boiso, Manuel
Hedenmalm, Håkan
Kaashoek, Marinus A.
Montes Rodríguez, Alfonso
Treil, Sergei
Departamento: Universidad de Sevilla. Departamento de Análisis Matemático
Fecha: 2014
Publicado en: Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation
ISBN/ISSN: 9783034806473
9783034806480
Tipo de documento: Capítulo de Libro
Resumen: In this survey we collect and discuss some recent results on the so called “Furstenberg set problem”, which in its classical form concerns the estimates of the Hausdorff dimension (dimH) of the sets in the Fα-class: for a given α ∈ (0, 1], a set E ⊆ R2 is in the Fα-class if for each e ∈ S there exists a unit line segment `e in the direction of e such that dimH(` ∩ E) ≥ α. For α = 1, this problem is essentially equivalent to the “Kakeya needle problem”. Define γ(α) = inf {dimH(E) : E ∈ Fα}. The best known results on γ(α) are the following inequalities: max {1/2 + α; 2α} ≤ γ(α) ≤ (1 + 3α)/2. In this work we approach this problem from a more general point of view, in terms of a generalized Hausdorff measure Hh associated with the dimension function h. We define the class Fh of Furstenberg sets associated to a given dimension function h. The natural requirement for a set E to belong to Fh, is that Hh(`e ∩ E) > 0 for each direction. We generalize the known results in terms of “logarithmi...
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Tamaño: 552.7Kb
Formato: PDF

URI: http://hdl.handle.net/11441/47721

DOI: 10.1007/978-3-0348-0648-0_27

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