K-divergent lattices

TL;DR

This paper introduces the concept of $k$-divergence in topological dynamics, explores its existence in lattice flows, and calculates the Hausdorff dimension of the set of $k$-divergent lattices, connecting to Diophantine approximation.

Contribution

It defines $k$-divergence, proves their existence for all $k\,\geq 0$, and applies parametric geometry of numbers to determine their Hausdorff dimension.

Findings

Existence of $k$-divergent lattices for all $k\geq 0

Calculation of Hausdorff dimension of $k$-divergent lattice sets

Connection between $k$-divergence and Diophantine approximation

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