Schmidt's game, badly approximable matrices and fractals

Abstract

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), then dim K = dim K ∩ BA ( M , N ) .