FURSTENBERG SETS FOR A FRACTAL SET OF DIRECTIONS

In this paper we study the behavior of the size of Furstenberg sets with respect to the size of the set of directions defining it. For any pair alpha, beta is an element of (0,1], we will say that a set E subset of R-2 is an F-alpha beta-set if there is a subset L of the unit circle of Hausdorff dimension at least beta and, for each direction e in L, there is a line segment l(e) in the direction of e such that the Hausdorff dimension of the set E boolean AND l(e) is equal to or greater than alpha. The problem is considered in the wider scenario of generalized Hausdorff measures, giving estimates on the appropriate dimension functions for each class of Furstenberg sets. As a corollary of our main results, we obtain that dim(E) >= max {alpha + beta/2; 2 alpha + beta - 1} for any E is an element of F-alpha beta. In particular we are able to extend previously known results to the "endpoint" alpha = 0 case.