Furstenberg sets for a fractal set of directions

Abstract:

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.