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Counting in Lattice Orbits

Alex Kontorovich, Christopher Lutsko

Abstract

Given a discrete lattice, $Γ< \text{SL}_m(\mathbb{R})$, and a base point $o\in \mathbb{R}^m$, let $N_Γ(T)$ denote the number of points in the orbit $o\cdot Γ$ whose (Euclidean) length is bounded by a growing parameter, $T$. We demonstrate an abstract spectral method capable of obtaining strong asymptotic estimates for $N_Γ(T)$ without relying on the meromorphic continuation of higher rank Langlands Eisenstein series.

Counting in Lattice Orbits

Abstract

Given a discrete lattice, , and a base point , let denote the number of points in the orbit whose (Euclidean) length is bounded by a growing parameter, . We demonstrate an abstract spectral method capable of obtaining strong asymptotic estimates for without relying on the meromorphic continuation of higher rank Langlands Eisenstein series.
Paper Structure (9 sections, 5 theorems, 61 equations)

This paper contains 9 sections, 5 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Plan of paper
  3. Preliminaries
  4. Group decomposition for $m=3$
  5. Group decomposition for general $m$
  6. Decomposition of $L^2(\Gamma \backslash G/K)$ into irreducibles and the spectral theorem
  7. Proof of Theorem \ref{['thm:m general']}
  8. Unfolding and the differential equation
  9. Proof of Theorem \ref{['thm:smooth m']}

Key Result

Theorem 1

There exist constants $c_0>0$, $c_1, \dots, c_k$ and $\eta_m>0$ such that

Theorems & Definitions (9)

  • Theorem 1
  • Remark
  • Remark
  • Theorem 2: Structure Theorem of the Haar measure and Casimir operator
  • proof
  • Theorem 3: Abstract Spectral Theorem
  • Theorem 4
  • proof : Proof of Theorem \ref{['thm:m general']} from Theorem \ref{['thm:smooth m']}
  • Theorem 5: Main Identity