Hausdorff dimension of Besicovitch sets of Cantor graphs

We consider the Hausdorff dimension of planar Besicovitch sets for rectifiable sets Gamma, that is, 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(2-s2,1s)$\min (2-s<^>2,\frac{1}{s})$, where s=dimC$s={\rm dim}\, C$.