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Abstract:

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.

Registro:

Documento: Artículo
Título:Small Furstenberg sets
Autor:Molter, U.; Rela, E.
Filiación:Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428 Capital Federal, Argentina
IMAS - CONICET, Argentina
Palabras clave:Dimension function; Furstenberg sets; Hausdorff dimension; Jarník's theorems
Año:2013
Volumen:400
Número:2
Página de inicio:475
Página de fin:486
DOI: http://dx.doi.org/10.1016/j.jmaa.2012.11.001
Título revista:Journal of Mathematical Analysis and Applications
Título revista abreviado:J. Math. Anal. Appl.
ISSN:0022247X
PDF:https://bibliotecadigital.exactas.uba.ar/download/paper/paper_0022247X_v400_n2_p475_Molter.pdf
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_0022247X_v400_n2_p475_Molter

Referencias:

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Citas:

---------- APA ----------
Molter, U. & Rela, E. (2013) . Small Furstenberg sets. Journal of Mathematical Analysis and Applications, 400(2), 475-486.
http://dx.doi.org/10.1016/j.jmaa.2012.11.001
---------- CHICAGO ----------
Molter, U., Rela, E. "Small Furstenberg sets" . Journal of Mathematical Analysis and Applications 400, no. 2 (2013) : 475-486.
http://dx.doi.org/10.1016/j.jmaa.2012.11.001
---------- MLA ----------
Molter, U., Rela, E. "Small Furstenberg sets" . Journal of Mathematical Analysis and Applications, vol. 400, no. 2, 2013, pp. 475-486.
http://dx.doi.org/10.1016/j.jmaa.2012.11.001
---------- VANCOUVER ----------
Molter, U., Rela, E. Small Furstenberg sets. J. Math. Anal. Appl. 2013;400(2):475-486.
http://dx.doi.org/10.1016/j.jmaa.2012.11.001