Small Furstenberg Sets

MOLTER, URSULA M.; RELA, EZEQUIEL

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 2013 vol. 400 p. 475 - 486

0022-247X

For $\alpha$ in $(0,1]$, a subset $E$ of $\RR$ is called \textit{Furstenberg set} of type $\alpha$ or $F_\alpha$-set if for each direction $e$ in the unit circle 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 greater or equal than $\alpha$. In this paper we use generalized Hausdorff measures to give estimates on the size of these sets. Our main result is to obtain a sharp dimension estimate for a whole class of {\em zero}-dimensional Furstenberg type sets. Namely, for $\h_\gamma(x)=\log^{-\gamma}(\frac{1}{x})$, $\gamma>0$, we construct a set $E_\gamma\in F_{\h_\gamma}$ of Hausdorff dimension not greater than $\frac{1}{2}$. Since in a previous work we showed that $\frac{1}{2}$ is a lower bound for the Hausdorff dimension of any $E\in F_{\h_\gamma}$, with the present construction, the value $\frac{1}{2}$ is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions $\h_\gamma$.