On the generalized Hausdorff dimension of Besicovitch sets

Keich (1999) showed that the sharp gauge function for the generalized Hausdorff dimension of Besicovitch sets in ${\mathbb{R}}^2$ is between $r^2\log 1\mathrm{/}r$ and $r^2(\log 1\mathrm{/}r)(\log \log 1\mathrm{/}r{)}^{2+\epsilon }$ by refining an argument of Bourgain (1991). It is not known whether the iterated logarithms in Keich’s bound are necessary. In this paper we construct a family of Besicovitch line sets whose sharp gauge function is smaller than $r^2(\log 1\mathrm{/}r)(\log \log 1\mathrm{/}r{)}^{\epsilon }$. Moreover, these Besicovitch sets are minimal in the sense that there is essentially only one line in the set pointing in each direction.