Small Furstenberg sets

For αα in (0,1](0,1], a subset EE of R2R2 is called a Furstenberg set   of type αα or FαFα-set if for each direction ee in the unit circle there is a line segment ℓeℓe in the direction of ee such that the Hausdorff dimension of the set E∩ℓeE∩ℓe is greater than or equal to αα. 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>0γ>0, we construct a set Eγ∈FhγEγ∈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∈FhγE∈Fhγ, with the present construction, the value 12 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions hγhγ.