Schmidt's game, badly approximable matrices and fractals

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