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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^mthe distance between x and the point {Aq} is at least c(x) |q|^{-m/n}. It isshown that the intersection of Bad_A with any suitably regular fractal set isof maximal Hausdorff dimension. The linear form systems investigated in thispaper are natural extensions of irrational rotations of the circle. Even in thelatter one-dimensional case, the results obtained are new.

On shrinking targets for Z^m actions on tori