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Effective equidistribution of expanding translates in the space of affine lattices

Wooyeon Kim

TL;DR

The paper proves a polynomially effective equidistribution result for expanding translates in the space of $d$-dimensional affine lattices, $Y=\operatorname{SL}_d(\mathbb{R})\ltimes \mathbb{R}^d/\operatorname{SL}_d(\mathbb{Z})\ltimes \mathbb{Z}^d$, for all $d\ge 2$. Its approach blends an effective base equidistribution on $X=\operatorname{SL}_d(\mathbb{R})/\operatorname{SL}_d(\mathbb{Z})$ with a fiber-wise Fourier analysis on the torus fiber $\mathbb{T}^d$, using a partition of unity on $X$ and a geometric decomposition of $Y$. A key innovation is a Diophantine-geometry mechanism that translates large nonzero Fourier coefficients into high-concentration scenarios on auxiliary measures, which is then ruled out by a Weyl-type argument together with an effective equidistribution on $X$. The main quantitative bound is expressed in terms of a Diophantine growth function $\zeta(b,T)$, yielding explicit polynomial rates depending on the Diophantine type of the torus coordinate $b$, and it extends to general diagonal subgroups. This advances effective measure rigidity in the affine-lattice setting and opens avenues for quantitative homogeneous dynamics in related semi-direct product actions.

Abstract

We prove a polynomially effective equidistribution result for expanding translates in the space of $d$-dimensional affine lattices for any $d\ge 2$.

Effective equidistribution of expanding translates in the space of affine lattices

TL;DR

The paper proves a polynomially effective equidistribution result for expanding translates in the space of -dimensional affine lattices, , for all . Its approach blends an effective base equidistribution on with a fiber-wise Fourier analysis on the torus fiber , using a partition of unity on and a geometric decomposition of . A key innovation is a Diophantine-geometry mechanism that translates large nonzero Fourier coefficients into high-concentration scenarios on auxiliary measures, which is then ruled out by a Weyl-type argument together with an effective equidistribution on . The main quantitative bound is expressed in terms of a Diophantine growth function , yielding explicit polynomial rates depending on the Diophantine type of the torus coordinate , and it extends to general diagonal subgroups. This advances effective measure rigidity in the affine-lattice setting and opens avenues for quantitative homogeneous dynamics in related semi-direct product actions.

Abstract

We prove a polynomially effective equidistribution result for expanding translates in the space of -dimensional affine lattices for any .

Paper Structure

This paper contains 25 sections, 30 theorems, 251 equations.

Table of Contents

  1. Introduction
  2. Backgrounds
  3. Main results
  4. Sketch of the proof of Theorem \ref{['mainthm']}
  5. Preliminaries
  6. Metrics and norms
  7. Compactness, height, and injectivity
  8. Sobolev norms
  9. Effective equidistribution in $X$ and measure estimates
  10. The Horospherical subgroup
  11. $HH^0H^-$ decomposition of $G$
  12. A fundamental domain of $X=\operatorname{SL}_d(\mathbb{R})/\operatorname{SL}_d(\mathbb{Z})$
  13. Parametrization of $Y$
  14. A partition of unity of $X$
  15. Dual affine lattices: reduction to the case $m\leq n$
  16. ...and 10 more sections

Key Result

Theorem 1.1

KM96 Let $V\subset H$ be a fixed neighborhood of the identity in $H$ with smooth boundary and compact closure. Then there exists a constant $\delta_0>0$ only depending on $m$ and $n$ so that the following holds. For any compact set $K\subset X$, there exists a constant $T(K)\ge 0$ such that for any $t\ge T(K)$, $f\in C_c^{\infty}(X)$, and $x\in K$. Here, $\mathcal{S}$ is a suitable Sobolev norm o

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 40 more