Schmidt’s Game, Badly Approximable Linear Forms and Fractals
نویسنده
چکیده
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 , 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 satisfying the open set condition, (the Cantor’s ternary set, Koch’s curve and Sierpinski’s gasket to name a few known examples), then dimK = dimK ∩BA(M,N).
منابع مشابه
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