Abstract: We study the set of badly approximable vectors in the setting of inhomogeneous Diophantine approximation. We show that, under certain conditions, this set of vectors has null intersection with affine rational lines. Our approach is based on dynamics in the space of grids and we explain how the mixing properties of sequences of group homomorphisms between tori comes into play. Based on ongoing work with Nikolay Moshchevitin and Uri Shapira.