Some measure-theoretical and dynamical properties of successive minima on the space of unimodular lattices

TL;DR

This paper investigates the relationship between lattice bases and successive minima, providing measure-theoretical estimates and exploring logarithm laws related to higher successive minima in the space of unimodular lattices.

Contribution

It introduces new measure-theoretical estimates for the distribution of successive minima and discusses associated logarithm laws in the context of unimodular lattices.

Findings

Derived estimates for the measure-theoretical distribution of successive minima

Established connections between lattice basis and successive minima

Explored logarithm laws related to higher successive minima

Abstract

In this article, we study the relation between lattice basis and successive minima and give an estimate for the measure-theoretical distribution of successive minima. As consequences, we also discuss some logarithm laws associated to higher sucessive minima.