Polynomial effective equidistribution

TL;DR

This paper establishes effective equidistribution results with polynomial error rates for unipotent orbits in certain arithmetic quotients of complex and real special linear groups, advancing understanding of their distribution properties.

Contribution

It provides the first effective polynomial error bounds for unipotent orbit equidistribution in these specific arithmetic quotients, using innovative techniques.

Findings

Proves polynomial effective equidistribution theorems.

Uses Margulis functions and incidence geometry tools.

Leverages spectral gap properties of the spaces.

Abstract

We prove effective equidistribution theorems, with polynomial error rate, for orbits of the unipotent subgroups of $\operatorname{SL}_2(\mathbb R)$ in arithmetic quotients of $\operatorname{SL}_2(\mathbb C)$ and $\operatorname{SL}_2(\mathbb R)\times\operatorname{SL}_2(\mathbb R)$. The proof is based on the use of a Margulis function, tools from incidence geometry, and the spectral gap of the ambient space.