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Effective equidistribution in rank 2 homogeneous spaces and values of quadratic forms

Elon Lindenstrauss, Amir Mohammadi, Zhiren Wang, Lei Yang

TL;DR

The paper develops polynomial-rate effective equidistribution results for one-parameter unipotent orbits in rank-2 quasi-split groups, combining energy methods, projection theorems, and exponential mixing to convert high-dimensional dynamical information into explicit distributional bounds. It then applies these results to the Oppenheim conjecture for indefinite ternary quadratic forms, delivering a quantitative, polynomial-rate proof and strengthening prior work by Eskin–Margulis–Mozes and Margulis. The core innovations are the modified-energy framework, the projection theorems of Gan–Guo–Wang, and a robust linearization/closing-lemma toolkit that handles large-divergence and cusp phenomena, enabling effective equidistribution and precise lattice-point counting results relevant to number theory. The combined dynamical and Diophantine analysis yields new quantitative insights into the distribution of quadratic form values and their relation to periodic and near-periodic homogeneous dynamics, with broader implications for extrapolating effective rates in arithmetic settings.

Abstract

We establish effective equidistribution theorems, with a polynomial error rate, for orbits of unipotent subgroups in quotients of quasi-split, almost simple Linear algebraic groups of absolute rank 2. As an application, inspired by the results of Eskin, Margulis and Mozes, we establish quantitative results regarding the distribution of values of an indefinite ternary quadratic form at integer points, giving in particular an effective and quantitative proof of the Oppenheim Conjecture.

Effective equidistribution in rank 2 homogeneous spaces and values of quadratic forms

TL;DR

The paper develops polynomial-rate effective equidistribution results for one-parameter unipotent orbits in rank-2 quasi-split groups, combining energy methods, projection theorems, and exponential mixing to convert high-dimensional dynamical information into explicit distributional bounds. It then applies these results to the Oppenheim conjecture for indefinite ternary quadratic forms, delivering a quantitative, polynomial-rate proof and strengthening prior work by Eskin–Margulis–Mozes and Margulis. The core innovations are the modified-energy framework, the projection theorems of Gan–Guo–Wang, and a robust linearization/closing-lemma toolkit that handles large-divergence and cusp phenomena, enabling effective equidistribution and precise lattice-point counting results relevant to number theory. The combined dynamical and Diophantine analysis yields new quantitative insights into the distribution of quadratic form values and their relation to periodic and near-periodic homogeneous dynamics, with broader implications for extrapolating effective rates in arithmetic settings.

Abstract

We establish effective equidistribution theorems, with a polynomial error rate, for orbits of unipotent subgroups in quotients of quasi-split, almost simple Linear algebraic groups of absolute rank 2. As an application, inspired by the results of Eskin, Margulis and Mozes, we establish quantitative results regarding the distribution of values of an indefinite ternary quadratic form at integer points, giving in particular an effective and quantitative proof of the Oppenheim Conjecture.

Paper Structure

This paper contains 38 sections, 67 theorems, 699 equations.

Table of Contents

  1. Effective equidistribution
  2. Applications to the Oppenheim conjecture
  3. Polynomially effective equidistribution theorems
  4. Notation and preliminary results
  5. The principal ${\rm{SL}}_2(\mathbb{R})$ in $G$
  6. Avoidance principles in homogeneous spaces
  7. Nondivergence results
  8. Inheritance of the Diophantine property
  9. Closing Lemma
  10. Mixing property and equidistribution
  11. Modified energy and projection theorems
  12. Proof of Theorem \ref{['thm: proj thm']}
  13. Linear algebra lemma and the energy
  14. Margulis functions and projection theorems
  15. The main proposition
  16. ...and 23 more sections

Key Result

Theorem 1.1

For every $x_0\in X$ and large enough $R$ (depending explicitly on $x_0$), for any $T \geq R^{a:main-3-1}$, at least one of the following holds. The constants ${A_{1}}$ and ${\kappa_{1}}$ are positive, and depend on $X$ but not on $x_0$.

Theorems & Definitions (119)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Lemma 3.2: LM-PolyDensity, Lemma 2.1
  • Lemma 3.3: LM-PolyDensity, Lemma 2.2
  • Lemma 3.4: LM-PolyDensity, Lemma 2.3
  • Lemma 3.5: cf. Lemma A.1, LMW22
  • Proposition 4.2
  • ...and 109 more