Schmidt's game, badly approximable matrices and fractals |
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Authors: | Lior Fishman |
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Institution: | MS 50, Mathematics Department, Brandeis University, PO Box 549110, Waltham, MA 02454, United States |
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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 RMN, 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 RMN 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). |
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Keywords: | Schmit's game Badly approximable matrices Open set condition Fractals Friendly measures Hausdorff measure Hausdorff dimension Winning dimension |
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