Schmidt's game, badly approximable matrices and fractals

We prove that for every M, N ∈ N, if τ is a Borel, finite, absolutely friendly measure supported on a compact subset K of R^{M N}, then K ∩ BA (M, N) is a winning set in Schmidt's game sense played on K, where BA (M, N) is the set of badly approximable M × N matrices. As an immediate consequence we have the following application. If K is the attractor of an irreducible finite family of contracting similarity maps of R^{M N} satisfying the open set condition (the Cantor's ternary set, Koch's curve and Sierpinski's gasket to name a few known examples), thendim K = dim K ∩ BA (M, N) .