Small Furstenberg sets

Abstract

For α in (0, 1], a subset E of R 2 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 ( 1 x ), 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.