k-divergent lattices

We introduce a novel concept in topological dynamics, referred to as kdivergence, which extends the notion of divergent orbits.Motivated by questions in the theory of inhomogeneous Diophantine approximations, we investigate this notion in the dynamical system given by a certain flow on the space of unimodular lattices in R d .Our main result is the existence of k-divergent lattices for any k ≥ 0. In fact, we utilize the emerging theory of parametric geometry of numbers and calculate the Hausdorff dimension of the set of k-divergent lattices.