Parametric Geometry of Numbers with General Flow

TL;DR

This paper extends the parametric geometry of numbers to general diagonal flows on lattices and computes the Hausdorff dimension of trajectories with specific behaviors under nonstandard metrics.

Contribution

It introduces a generalization of parametric geometry of numbers to broader diagonal flows and calculates the Hausdorff dimension of certain trajectory sets.

Findings

Extended parametric geometry of numbers to general diagonal flows

Computed Hausdorff dimension of trajectory sets under nonstandard metrics

Provided bounds on Hausdorff dimension in standard metrics

Abstract

We extend results of parametric geometry of numbers to a general diagonal flow on the space of lattices. Moreover, we compute the Hausdorff dimension of the set of trajectories with every given behavior, with respect to a nonstandard metric and thereby attain bounds on the standard ones.