Small Furstenberg sets

• Published in: Journal of mathematical analysis and applications Vol. 400; no. 2; pp. 475 - 486
• Main Authors: Molter, Ursula, Rela, Ezequiel
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.04.2013
• Subjects: Dimension function; Furstenberg sets; Hausdorff dimension; Jarník’s theorems; Furstenberg sets; Dimension function; Hausdorff dimension; Jarník’s theorems
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2012.11.001

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 functions hγ.