  Keich (1999) showed that the sharp gauge function for the generalized
Hausdorff dimension of Besicovitch sets in $\mathbb R^2$ is between $r^2\log
1/r$ and $r^2(\log 1/r) (\log\log 1/r)^{2+\varepsilon}$ 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 $r^2(\log 1/r) (\log\log
1/r)^{\varepsilon}$. Moreover, these Besicovitch sets are minimal in the sense
that there is essentially only one line in the set pointing in each direction.
