Polynomially effective equidistribution for certain unipotent subgroups in quotients of perfect Lie groups

TL;DR

This paper establishes an effective equidistribution theorem with polynomial error for certain unipotent subgroup orbits in arithmetic quotients of perfect Lie groups, advancing understanding of orbit distribution and quadratic form values.

Contribution

It introduces the first effective polynomial error equidistribution results for non-horospherical unipotent subgroups in semisimple quotients, using novel techniques like a sub-modularity inequality.

Findings

Effective equidistribution with polynomial error rate for specific unipotent subgroups.

First such results for non-horospherical unipotent subgroups in semisimple quotients.

Polynomial error estimates for the distribution of lattice orbits and the Oppenheim conjecture.

Abstract

We prove an effective equidistribution theorem for orbits of certain unipotent subgroups in arithmetic quotients of perfect Lie groups with a polynomial error term. Even for semisimple quotients, our result provides the first infinite family of examples where effective equidistribution with polynomial error rate is obtained for non-horospherical unipotent subgroups. The proof is based on the spectral gap of the ambient space, an effective closing lemma, Bourgain's discretized projection theorem, and a sub-modularity inequality in irreducible representation. The sub-modularity inequality is crucial to our proof and is of independent interest. As applications, we obtain effective estimates on distribution of lattice orbits on homogeneous spaces, as well as an effective version of the Oppenheim conjecture for indefinite quadratic forms with a polynomial error rate in all dimension $d \geq 3$.