Small Furstenberg sets

Details Small Furstenberg sets Small Furstenberg sets

2010-06-24

arxiv.org/abs/1006.4862v2

Mathematics

Classical Analysis and ODEs

Completely revised and rewritten

Scientific paper

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 show that if $\alpha > 0$, there exists a set $E\in F_\alpha$ such that $\HH{g}(E)=0$ for $g(x)=x^{1/2+3/2\alpha}\log^{-\theta}(\frac{1}{x})$, $\theta>\frac{1+3\alpha}{2}$, which improves on the the previously known bound, that $H^{\beta}(E) = 0$ for $\beta>1/2+3/2\alpha$. Further, by refining the argument in a subtle way, we are able 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 1/2. Since in a previous work we showed that 1/2 is a lower bound for the Hausdorff dimension of any $E\in F_{\h_\gamma}$, with the present construction, the value 1/2 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions $\h_\gamma$.

Affiliated with

Also associated with

No associations

Rating

Small Furstenberg sets does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Small Furstenberg sets, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Small Furstenberg sets will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-676594