Let $A$ be an $n \times m$ matrix with real entries. Consider the set $\mathbf{Bad}_A$ of $\mathbf{x} \in [0,1)^n$ for which there exists a constant $c(\mathbf{x})>0$ such that for any $\mathbf{q} \in \mathbb{Z}^m$ the distance between $\mathbf{x}$ and the point $\{A \mathbf{q}\}$ is at least $c(\mathbf{x}) |\mathbf{q}|^{-m/n}$. It is shown that the intersection of $\mathbf{Bad}_A$ 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.