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Small Furstenberg sets
For in , a subset of \RR is called Furstenberg set of
type or -set if for each direction in the unit circle
there is a line segment in the direction of such that the
Hausdorff dimension of the set is greater or equal than .
In this paper we show that if , there exists a set
such that \HH{g}(E)=0 for
,
, which improves on the the previously known bound,
that for . 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_\gamma(x)=\log^{-\gamma}(\frac{1}{x}), , 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.Comment: Final versio
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