Schmidt's game, Badly Approximable Linear Forms and Fractals

TL;DR

This paper demonstrates that the set of badly approximable matrices intersects certain fractal sets in a winning way under Schmidt's game, implying these intersections have full Hausdorff dimension.

Contribution

It establishes that for any compact fractal set with a friendly measure, the intersection with badly approximable matrices is winning, extending Schmidt's game theory to fractals.

Findings

The intersection of fractals with badly approximable matrices is winning in Schmidt's game.

Such intersections have full Hausdorff dimension equal to the fractal's dimension.

Results apply to classic fractals like the Cantor set, Koch curve, and Sierpinski gasket.

Abstract

We prove that for every two natural numbers M and N, if Tau is a Borel, finite, absolutely friendly measure on a compact set K of R^MN, then the intersection of K and 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\times 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\times N) satisfying the open set condition, (the Cantor ternary set, Koch's curve and Sierpinski's gasket to name a few examples), the dimK=dimK\capBA(M,N).