On shrinking targets for Z^m actions on tori

Let A be an n by m matrix with real entries. Consider the set Bad_A of x \in [0,1)^n for which there exists a constant c(x)>0 such that for any q \in Z^m the distance between x and the point {Aq} is at least c(x) |q|^{-m/n}. It is shown that the intersection of Bad_A with any suitably regular fractal set is of maximal Hausdorff dimension. The linear form systems investigated in this paper are natural extensions of irrational rotations of the circle. Even in the latter one-dimensional case, the results obtained are new.