Furstenberg sets for a fractal set of directions

MOLTER, URSULA; RELA, EZEQUIEL

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY

AMER MATHEMATICAL SOC

Lugar: Providence; Año: 2012 vol. 140 p. 2753 - 2765

0002-9939

In this note 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 $alpha,etain(0,1]$, we will say that a set $Esubset R^2$ is an $F_{alphaeta}$-set if there is a subset $L$ of the unit circle of Hausdorff dimension at least $eta$ 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 $Ecapell_e$ is equal 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 $dim(E)gemaxleft{alpha+rac{eta}{2} ; 2alpha+eta -1 ight}$ for any $Ein F_{alphaeta}$. In particular we are able to extend previously known results to the ``endpoint´´ $alpha=0$ case.