INVESTIGADORES
MOLTER ursula Maria
artículos
Título:
Small Furstenberg Sets
Autor/es:
MOLTER, URSULA M.; RELA, EZEQUIEL
Revista:
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
Editorial:
ACADEMIC PRESS INC ELSEVIER SCIENCE
Referencias:
Lugar: Amsterdam; Año: 2013 vol. 400 p. 475 - 486
ISSN:
0022-247X
Resumen:
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 $\h_\gamma$.