Polynomially effective equidistribution for certain unipotent subgroups in quotients of semisimple Lie groups
Zuo Lin
TL;DR
The paper develops a polynomial-rate effective equidistribution theory for orbits of non-horospherical Ad-unipotent subgroups in arithmetic quotients of semisimple Lie groups. The approach combines a multi-stage strategy: an effective closing lemma to bootstrap an initial dimension, a subcritical/projection framework to improve transverse dimension via irreducible representations and a robust Margulis-function analysis, and a Hölder–Brascamp–Lieb framework to convert projection control into entropy bounds. This yields polynomial error terms in equidistribution, with applications to lattice-orbit distribution and an effective Oppenheim conjecture in all dimensions d≥3. A key auxiliary result is a submodularity-type inequality for irreducible representations that feeds into the dimension-improvement machinery. Overall, the work advances effective homogeneous dynamics for broader unipotent subgroups and provides tools potentially useful for quantitative number-theoretic applications and spectral-gap style analyses in semisimple quotients.
Abstract
We prove an effective equidistribution theorem for orbits of certain unipotent subgroups in arithmetic quotients of semisimple Lie groups with a polynomial error term. This provides the first infinite family of examples where effective equidistribution, with polynomial error rate, of non-horospherical unipotent subgroups in semisimple quotients is obtained. As applications, we obtain effective estimate 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 $d \geq 3$. We also prove a sub-modularity inequality for irreducible representation of connected algebraic group, which is crucial to our proof and is of independent interest.