K-divergent lattices

We introduce a novel concept in topological dynamics, referred to as \(k\)-divergence, 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 \(\mathbb{R}^d\). Our main result is the existence of \(k\)-divergent lattices for any \(k\geq 0\). In fact, we utilize the emerging theory of parametric geometry of numbers and calculate the Hausdorff dimension of the set of \(k\)-divergent lattices.