Furstenberg sets for a fractal set of directions

In this note we study the behavior of the size of Fursten- berg sets with respect to the size of the set of directions dening it. For any pair �;� 2 (0;1), we will say that a set ER 2 is an F��-set if there is a subset L of the unit circle of Hausdordimension at least � and, for each direction e in L, there is a line segmente in the direc- tion of e such that the Hausdordimension of the set E \ `e is equal or greater than �. The problem is considered in the wider scenario of generalized Hausdormeasures, giving estimates on the appropriate di- mension functions for each class of Furstenberg sets. As a corollary of our main results, we obtain that dim(E) � max � � + � ;2� + � 1 � for any E 2 F��. In particular we are able to extend previously known results to the \endpoint" � = 0 case.