K-divergent lattices

Abstract

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 $k\ge 0$. In fact, we utilize the emerging theory of parametric geometry of numbers and calculate the Hausdorff dimension of the set of $k$-divergent lattices.