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 $\alpha ,\beta \in (0,1]$, we will say that a set $E\subset \mathbb{R}^2$ 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 $\ell _e$ in the direction of $e$ such that the Hausdorff dimension of the set $E\cap \ell _e$ 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 $\dim (E)\ge \max \left \{\alpha +\frac {\beta }{2} ; 2\alpha +\beta -1\right \}$ for any $E\in F_{\alpha \beta }$. In particular we are able to extend previously known results to the ``endpoint'' $\alpha =0$ case.