Quadratic forms of signature $(2, 2)$ or $(3, 1)$ I: effective equidistribution in quotients of $\mathrm{SL}_4(\mathbb{R})$

TL;DR

This paper establishes an effective equidistribution theorem for certain orbits in quotients of SL(4,R), providing polynomial error bounds, and sets the stage for proving an effective Oppenheim conjecture for specific indefinite quadratic forms.

Contribution

It introduces an effective equidistribution result for horospherical orbits in quotients of SL(4,R) with polynomial error terms, advancing the understanding of dynamics related to quadratic forms.

Findings

Proves effective equidistribution with polynomial error bounds.

Applies to orbits of horospherical subgroups in SO(2,2) and SO(3,1).

Lays groundwork for an effective Oppenheim conjecture proof.

Abstract

We prove an effective equidistribution theorem for orbits of horospherical subgroups of $\mathrm{SO}(2, 2)$ and $\mathrm{SO}(3, 1)$ in quotients of $\mathrm{SL}_4(\mathbb{R})$ with a polynomial error term. In a forthcoming paper, we will use this theorem to prove an effective version of the Oppenheim conjecture for indefinite quadratic forms of signature $(2, 2)$ or $(3, 1)$ with a polynomial error rate.