Small Furstenberg sets

For α in ( 0 , 1 ] , a subset E of R 2 is called a Furstenberg set of type α or F α -set if for each direction e in the unit circle there is a line segment ℓ e in the direction of e such that the Hausdorff dimension of the set E ∩ ℓ e 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 h γ ( x ) = log − γ ( 1 x ) , γ > 0 , we construct a set E γ ∈ F h γ 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 ∈ F h γ , with the present construction, the value 1 2 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions  h γ .