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Improving dimension estimates for Furstenberg-type sets

Opened Access Improving dimension estimates for Furstenberg-type sets


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Autor: Molter, Úrsula María
Rela, Ezequiel
Departamento: Universidad de Sevilla. Departamento de Análisis Matemático
Fecha: 2010-01-30
Publicado en: Advances in Mathematics, 223 (2), 672-688.
Tipo de documento: Artículo
Resumen: In this paper we study the problem of estimating the generalized Hausdorff dimension of Furstenberg sets in the plane. For α∈(0,1], a set F in the plane is said to be an α-Furstenberg set if for each direction e there is a line segment ℓe in the direction of e for which dimH(ℓe∩F)⩾α. It is well known that , and it is also known that these sets can have zero measure at their critical dimension. By looking at general Hausdorff measures Hh defined for doubling functions, that need not be power laws, we obtain finer estimates for the size of the more general h-Furstenberg sets. Further, this approach allow us to sharpen the known bounds on the dimension of classical Furstenberg sets. The main difficulty we had to overcome, was that if Hh(F)=0, there always exists g≺h such that Hg(F)=0 (here ≺ refers to the natural ordering on general Hausdorff dimension functions). Hence, in order to estimate the measure of general Furstenberg sets, we have to consider dimension functions that are a true...
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Cita: Molter, Ú.M. y Rela, E. (2010). Improving dimension estimates for Furstenberg-type sets. Advances in Mathematics, 223 (2), 672-688.
Tamaño: 229.3Kb
Formato: PDF


DOI: 10.1016/j.aim.2009.08.019

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