Schmidt’s game, fractals, and orbits of toral endomorphisms

Abstract

Abstract Given an integer matrix M ∈GL n (ℝ) and a point y ∈ℝ n /ℤ n , consider the set S. G. Dani showed in 1988 that whenever M is semisimple and y ∈ℚ n /ℤ n , the set $ \tilde E(M,y)$ has full Hausdorff dimension. In this paper we strengthen this result, extending it to arbitrary M ∈GL n (ℝ)∩M n × n (ℤ) and y ∈ℝ n /ℤ n , and in fact replacing the sequence of powers of M by any lacunary sequence of (not necessarily integer) m × n matrices. Furthermore, we show that sets of the form $ \tilde E(M,y)$ and their generalizations always intersect with ‘sufficiently regular’ fractal subsets of ℝ n . As an application, we give an alternative proof of a recent result [M. Einsiedler and J. Tseng. Badly approximable systems of affine forms, fractals, and Schmidt games. Preprint , arXiv:0912.2445] on badly approximable systems of affine forms.