Mathematics – Classical Analysis and ODEs
Scientific paper
2010-09-02
Mathematics
Classical Analysis and ODEs
13 pages
Scientific paper
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 $\alpha,\beta\in(0,1]$, we will say that a set $E\subset \R^2$ 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 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.
Molter Ursula
Rela Ezequiel
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