Small Furstenberg sets

• Autores: Molter, U.; Rela, E.
• Año de publicación: 2013
• Idioma: inglés
• Tipo de recurso: artículo
• Fuente: J. Math. Anal. Appl. 2013;400(2):475-486
• Materia: Dimension function; Furstenberg sets; Hausdorff dimension; Jarník's theorems
• Institución: Universidad Nacional de Buenos Aires. Facultad de Ciencias Exactas y Naturales

For α in (0, 1], a subset E of R2 is called a Furstenberg set of type α or Fα-set if for each direction e in the unit circle there is a line segment ℓe in the direction of e such that the Hausdorff dimension of the set E∩ℓe 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 12. Since in a previous work we showed that 12 is a lower bound for the Hausdorff dimension of any E∈Fhγ, with the present construction, the value 12 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functionshγ. © 2012 Elsevier Ltd.
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.