  We consider the Hausdorff dimension of planar Besicovitch sets for
rectifiable sets $\Gamma$, i.e. sets that contain a rotated copy of $\Gamma$ in
each direction. We show that for a large class of Cantor sets $C$ and
Cantor-graphs $\Gamma$ built on $C$, the Hausdorff dimension of any
$\Gamma$-Besicovitch set must be at least $\min\left(2-s^2,\frac{1}{s}\right)$,
where $s=\dim C$.
