Hausdorff dimension of Besicovitch sets of Cantor graphs

Abstract: A Besicovitch set is a set of points in Euclidean space which contains a unit line segment in every direction. A Kakeya needle set is a Besicovitch set in the plane with a stronger property, that a unit line segment can be rotated continuously through 180 degrees within it. It is a well-known result that in ℝ² the Hausdorff Dimension of the Besicovitch set is 2. We replace the straight line with a graph Γ built on a self-similar Cantor set C of dimension 1/2 and define B ⊂ ℝ² to be a Γ-Besicovitch set if B contains a rotated (and translated) copy of Γ in every direction. We will see that the Hausdorff Dimension of such a Γ-Besicovitch set is at least 7/4.