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

Given an integer matrix M ∈ GL_n(ℝ) and a point y ∈ ℝⁿ/ℤⁿ, consider the set Ẽ(M,y) ^def={x ∈ ℝⁿ : y ∉ {M^kx mod ℤⁿ : k ∈ ℕ}}̄. S. G. Dani showed in 1988 that whenever M is semisimple and y ∈ ℚⁿ/ℤⁿ, the set Ẽ(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 ∈ ℝⁿ/ℤⁿ, 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 Ẽ(M,y) and their generalizations always intersect with 'sufficiently regular' fractal subsets of ℝⁿ. 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.