Opened Access Small Furstenberg sets

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Author: Molter, Úrsula María
Rela, Ezequiel
Department: Universidad de Sevilla. Departamento de Análisis Matemático
Date: 2013-04-15
Published in: Journal of Mathematical Analysis and Applications, 400 (2), 475-486.
Document type: Article
Abstract: For α in (0, 1], a subset E of R2 is called Furstenberg set of type α or Fα-set if for each direction e in the unit circle there is a line segment `e in the direction of e such that the Hausdorff dimension of the set E ∩`e is greater than or equal to α. In this paper we use generalized Hausdorff measures to give estimates on the size of these sets. Our main result is to obtain a sharp dimension estimate for a whole class of zero-dimensional Furstenberg type sets. Namely, for hγ(x) = log−γ (1x), γ > 0, we construct a set Eγ ∈ Fhγ of Hausdorff dimension not greater than 1/2. Since in a previous work we showed that 1/2 is a lower bound for the Hausdorff dimension of any E ∈ Fhγ, with the present construction, the value 1/2 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions hγ.
Cite: Molter, Ú.M. y Rela, E. (2013). Small Furstenberg sets. Journal of Mathematical Analysis and Applications, 400 (2), 475-486.
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DOI: 10.1016/j.jmaa.2012.11.001

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