Small Furstenberg sets

Abstract

For $\alpha$ in $(0,1]$, a subset $E$ of $\RR$ is called 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 zero-dimensional Furstenberg type sets. Namely, for $\h_γ(x)=\log^{-γ}(\frac{1}{x})$, $\gamma >0$, we construct a set $E_γ\in F_{\h_γ}$ 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 $E\in F_{\h_γ}$, with the present construction, the value 1/2 is sharp for the whole class of Furstenberg sets associated to the zero dimensional functions $\h_γ$.

Final version

References in corpus (4)

• From rotating needles to stability of waves; emerging connections between combinatorics, analysis and PDE
• Some connections between Falconer's distance set conjecture, and sets of Furstenburg type
• Classical metric Diophantine approximation revisited
• Recent progress on the Kakeya conjecture