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Small Furstenberg sets

For $α$ in $(0,1]$, a subset $E$ of $\RR$ is called Furstenberg set of type $α$ or $F_α$-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 $α$. In this paper we show that if $α> 0$, there exists a set $E\in F_α$ such that $\HH{g}(E)=0$ for $g(x)=x^{1/2+3/2α}\log^{-θ}(\frac{1}{x})$, $θ>\frac{1+3α}{2}$, which improves on the the previously known bound, that $H^β(E) = 0$ for $β>1/2+3/2α$. 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 $\h_γ(x)=\log^{-γ}(\frac{1}{x})$, $γ>0$, 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 $\h_γ$.