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Polynomially effective equidistribution for unipotent orbits in products of $\mathrm{SL}_2$ factors

Elon Lindenstrauss, Amir Mohammadi, Lei Yang

TL;DR

The paper proves a polynomial-rate effective equidistribution result for one-parameter unipotent flows on S-arithmetic quotients associated with K-forms of SL_2^n, addressing obstructions from intermediate subgroups. It introduces a Bourgain-type projection theorem in the presence of obstructions and develops a global obstruction analysis to either force a measurable obstruction or achieve near-full dimension, which then yields equidistribution via spectral gap arguments. The approach proceeds in three phases—initial dimension via a closing lemma, dimension improvement through projection theory, and final equidistribution via exponential mixing—generalizing previous work and handling obstructions in a unified framework. This advances understanding of effective Ratner-type phenomena in higher-rank, product settings and strengthens connections to multidimensional projection and sum-product phenomena in both archimedean and non-archimedean contexts.

Abstract

We sketch the proof of an effective equidistribution theorem for one-parameter unipotent subgroups in $S$-arithmetic quotients arising from $\mathbf K$-forms of $\mathrm{SL}_2^{\mathsf n}$ where $\mathbf K$ is a number field. This gives an effective version of equidistribution results of Ratner and Shah with a polynomial rate. The key new phenomenon is the existence of many intermediate groups between the $\mathrm{SL}_2$ containing our unipotent and the ambient group, which introduces potential local and global obstruction to equidistribution. Our approach relies on a Bourgain-type projection theorem in the presence of obstructions, together with a careful analysis of these obstructions.

Polynomially effective equidistribution for unipotent orbits in products of $\mathrm{SL}_2$ factors

TL;DR

The paper proves a polynomial-rate effective equidistribution result for one-parameter unipotent flows on S-arithmetic quotients associated with K-forms of SL_2^n, addressing obstructions from intermediate subgroups. It introduces a Bourgain-type projection theorem in the presence of obstructions and develops a global obstruction analysis to either force a measurable obstruction or achieve near-full dimension, which then yields equidistribution via spectral gap arguments. The approach proceeds in three phases—initial dimension via a closing lemma, dimension improvement through projection theory, and final equidistribution via exponential mixing—generalizing previous work and handling obstructions in a unified framework. This advances understanding of effective Ratner-type phenomena in higher-rank, product settings and strengthens connections to multidimensional projection and sum-product phenomena in both archimedean and non-archimedean contexts.

Abstract

We sketch the proof of an effective equidistribution theorem for one-parameter unipotent subgroups in -arithmetic quotients arising from -forms of where is a number field. This gives an effective version of equidistribution results of Ratner and Shah with a polynomial rate. The key new phenomenon is the existence of many intermediate groups between the containing our unipotent and the ambient group, which introduces potential local and global obstruction to equidistribution. Our approach relies on a Bourgain-type projection theorem in the presence of obstructions, together with a careful analysis of these obstructions.
Paper Structure (6 sections, 3 theorems, 27 equations)

This paper contains 6 sections, 3 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. A Bourgain-type projection theorem
  3. Proof of Theorem \ref{['thm: main']}
  4. Initial dimension and a closing lemma
  5. Improving the dimension and obstructions
  6. From high dimension to equidistribution

Key Result

Theorem 1.1

For every $x_0\in X$ and large enough $R$ (depending logarithmically on the injectivity radius at $x_0$), for any $T \geq a:main R$, 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 (4)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Definition 3.3