Abstract: Consider a homogeneous space X - such as the space of lattices or the space of grids in Euclidean space - and the action of a diagonal one-parameter subgroup. In 1996 Kleinbock-Margulis proved that the set of points in X with precompact forward orbit has full Hausdorff dimension. We study a slight refinement of this problem: given some x in X, does the same dimension result hold for points with precompact forward orbit accumulating on x? We prove that, barring a certain topological condition on x, the codimension of this set is less than the dimension of the centralizer of the subgroup. All of this is inspired by a 1969 result of Davenport-Schmidt regarding badly approximable matrices and we explain this along with some new applications to Diophantine approximation. Joint work with Manfred Einsiedler and Dmitry Kleinbock.