Effective equidistribution for some one parameter unipotent flows

TL;DR

This paper establishes effective equidistribution results with polynomial error rates for unipotent flows in certain arithmetic quotients of complex and real special linear groups, using advanced geometric and spectral techniques.

Contribution

It introduces new effective equidistribution theorems for specific unipotent flows, leveraging a Margulis function, incidence geometry, and spectral gap methods.

Findings

Proves polynomial error rate equidistribution theorems

Applies techniques to arithmetic quotients of SL(2,C) and SL(2,R)

Utilizes spectral gap and incidence geometry tools

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.