A note on Diophatine approximation in $\rm{SL}_2(\mathbb{R})$

TL;DR

This paper provides a quantitative analysis of unipotent flow orbits in the space of lattices in SL(2,R), demonstrating their density with explicit bounds using Weil's bounds for Kloosterman sums.

Contribution

It introduces a quantitative version of the density of unipotent flow orbits in SL(2,R)/SL(2,Z), employing bounds from analytic number theory.

Findings

Unipotent flow orbits are dense in the lattice space.

Quantitative bounds are established using Weil's bounds for Kloosterman sums.

The results give explicit measures of orbit distribution.

Abstract

We prove a quantitative version of the following statement: the unipotent flow orbit of a typical lattice in $\rm{SL}_2(\mathbb{R})/\rm{SL}_2(\mathbb{Z})$ is dense. Our quantitative result uses A. Weil's bounds for Kloostermann sums.