Effective version of Ratner's equidistribution theorem for $\mathrm{SL}(3,\mathbb{R})$

TL;DR

This paper establishes an effective version of Ratner's equidistribution theorem for unipotent orbits in the space of 3x3 real matrices modulo integers, under a Diophantine condition, providing quantitative convergence results.

Contribution

It provides the first effective bounds for Ratner's theorem in the case of SL(3,R), advancing quantitative understanding of unipotent orbit distributions.

Findings

Derived explicit error bounds for equidistribution

Extended Ratner's theorem to effective versions in SL(3,R)

Applied Diophantine conditions to quantify orbit distribution

Abstract

In this paper, we will prove an effective version of Ratner's equidistribution theorem for unipotent orbits in $\mathrm{SL}(3,\mathbb{R})/\mathrm{SL}(3,\mathbb{Z})$ with a natural Diophantine condition.