Schmidt’s Game, Fractals, and Orbits of Toral Endomorphisms

Given an integer matrix M ∈ GLn(R) and a point y ∈ R/Z, consider the set Ẽ(M,y) def = { x ∈ R : y / ∈ {Mkx mod Zn : k ∈ N} } . S.G. Dani showed in 1988 that whenever M is semisimple and y ∈ Q/Z, the set Ẽ(M,y) is winning in the sense of W. Schmidt (a property implying density and full Hausdorff dimension). In this paper we strengthen this result, extending it to arbitrary y ∈ R/Z and M ∈ GLn(R) ∩Mn×n(Z), 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 R. As an application we strengthen recent results of [2, 22, 32] on badly approximable systems of affine forms.