Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Effective equidistribution of translates of tori in arithmetic homogeneous spaces and applications

Pratyush Sarkar

TL;DR

The paper proves effective equidistribution of large translates of tori in arithmetic homogeneous spaces $X=G/\Gamma$ for rank-$\le 2$ groups, obtaining power-saving error terms and applying these to count integral $3\times3$ matrices with a fixed characteristic polynomial. The authors develop a framework linking torus-equidistribution to unipotent dynamics via limiting nilpotent Lie algebras, and introduce the star-centralizing/centralizing concepts to control the input needed for general $G$. A key technical advance is the quantitative analysis of limiting nilpotent Lie algebras and the derivation of effective equidistribution for growing unipotent balls, together with quantitative non-divergence for translates of tori. The approach unifies Lie-theoretic nilpotent-element analysis with ergodic/dynamical methods and uses inputs from recent work in the area (LMW/LMWY) to achieve unconditional results in the $n=3$ setting, including explicit counting formulas and constants in favorable arithmetic cases. The results have significant implications for asymptotic lattice-point counts in spaces of matrices with prescribed invariants, illustrating a powerful method to derive effective counting from dynamical equidistribution in arithmetic homogeneous spaces.

Abstract

Let $Γ< G$ be an arithmetic lattice in a noncompact connected semisimple real algebraic group. For many such $G$ of rank at most $2$, in particular $G = \operatorname{SL}_3(\mathbb R)$, we prove effective equidistribution of large translates of tori in $G/Γ$. As an application, we obtain an asymptotic counting formula with a power saving error term for integral $3 \times 3$ matrices with a specified characteristic polynomial. These effectivize celebrated theorems of Eskin-Mozes-Shah.

Effective equidistribution of translates of tori in arithmetic homogeneous spaces and applications

TL;DR

The paper proves effective equidistribution of large translates of tori in arithmetic homogeneous spaces for rank- groups, obtaining power-saving error terms and applying these to count integral matrices with a fixed characteristic polynomial. The authors develop a framework linking torus-equidistribution to unipotent dynamics via limiting nilpotent Lie algebras, and introduce the star-centralizing/centralizing concepts to control the input needed for general . A key technical advance is the quantitative analysis of limiting nilpotent Lie algebras and the derivation of effective equidistribution for growing unipotent balls, together with quantitative non-divergence for translates of tori. The approach unifies Lie-theoretic nilpotent-element analysis with ergodic/dynamical methods and uses inputs from recent work in the area (LMW/LMWY) to achieve unconditional results in the setting, including explicit counting formulas and constants in favorable arithmetic cases. The results have significant implications for asymptotic lattice-point counts in spaces of matrices with prescribed invariants, illustrating a powerful method to derive effective counting from dynamical equidistribution in arithmetic homogeneous spaces.

Abstract

Let be an arithmetic lattice in a noncompact connected semisimple real algebraic group. For many such of rank at most , in particular , we prove effective equidistribution of large translates of tori in . As an application, we obtain an asymptotic counting formula with a power saving error term for integral matrices with a specified characteristic polynomial. These effectivize celebrated theorems of Eskin-Mozes-Shah.

Paper Structure

This paper contains 23 sections, 46 theorems, 235 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results
  3. Historical background
  4. Outline of the proofs of Theorems \ref{['thm:MainCounting']} and \ref{['thm:EquidistributionOfToriSL_3']}
  5. Organization
  6. Notation and preliminaries
  7. Big O, Ω, and Vinogradov notations
  8. Algebraic/Lie groups and Lie algebras
  9. Metrics and measures
  10. Height and injectivity radius
  11. Nilpotent elements and their centralizers
  12. Lie theoretic estimates for regular nilpotent elements
  13. Limiting nilpotent Lie algebras for regular nilpotent elements
  14. Limiting vector spaces
  15. Limiting nilpotent Lie algebras
  16. ...and 8 more sections

Key Result

Theorem 1.1

There exists $c_p > 0$ (depending only on $\|\boldsymbol{\cdot}\|$) and $\kappa > 0$ such that for all $T > 0$, we have

Figures (1)

  • Figure 1: In general, the weight space decomposition $\mathfrak{g} = \bigoplus_{j \in \mathscr{J}} \bigoplus_{k = -\varkappa_j}^{\varkappa_j} \mathcal{V}_j(k)$ can be represented by a diagram of the above fashion. As an example, the provided diagram is for $\mathfrak{g} := \mathop{\mathrm{\mathfrak{so}}}\nolimits(5, 2)$ and a regular nilpotent element $\mathsf{n} \in \mathfrak{g}$. Each row represents an irreducible representation of $\mathop{\mathrm{\mathfrak{sl}}}\nolimits_2(\mathsf{n})$ in $\mathfrak{g}$. In each row, each dot represents a weight space (which is $1$-dimensional) of $\mathop{\mathrm{\mathfrak{sl}}}\nolimits_2(\mathsf{n})$ in increasing order according to weights. Therefore, the center column represents $\mathfrak{g}(0) = \mathfrak{g}_0 = \mathfrak{a} \oplus \mathfrak{m}$ where the magenta dots form $\mathfrak{a}$ and the light green dots form $\mathfrak{m}$. The dots enclosed by the blue line are the highest weights and they form the centralizer $Z_\mathfrak{g}(\mathsf{n})$.

Theorems & Definitions (116)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • proof
  • Remark 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Remark 1.9
  • ...and 106 more