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Limit distributions for $\text{SO}(n,1)$ action on $k$-lattices in $\mathbb{R}^{n+1}$

Michael Bersudsky, Nimish A. Shah

TL;DR

This paper analyzes the asymptotic distribution of norm-ball averages for orbits of a lattice $\Gamma \subset \mathrm{SO}(n,1)^\circ$ acting on the moduli space $X_{r,n+1}$ of pairs of orthogonal lattices in $\mathbb{R}^{n+1}$. By combining a sums-to-integrals framework, explicit volume asymptotics, and equidistribution results derived from Ratner-type theory and Margulis’ banana trick, the authors identify a limiting semi-invariant measure concentrated on degenerate pairs lying tangent to the light cone, with a precise base-fiber decomposition over degenerate Grassmannian pairs. They treat both nondegenerate and degenerate cases, including a detailed analysis of the special two-dimensional degeneracies in low dimensions, which resolves the Sargent–Shapira conjecture in a special instance. In higher dimensions, they obtain the joint limiting distribution for pairs of orthogonal lattices and provide a complete framework distinguishing commensurable versus non-commensurable scenarios, thereby advancing the understanding of lattice distributions in homogeneous dynamics. The results connect to Furstenberg-type questions on Diophantine approximation and demonstrate a robust methodology for deriving limiting measures via volume estimates and unipotent invariance, with potential applications to broader lattice counting problems in semi-simple Lie groups.

Abstract

We study the asymptotic distribution of norm ball averages along orbits of a lattice $Γ\subset \text{SO}(n,1)$ acting on the moduli space of pairs of orthogonal discrete subgroups of $\mathbb{R}^{n+1}$ up to homothety. Our main result shows that, except for special $2$-lattices in $\mathbb{R}^3$ lying in hyperplanes tangent to the light cone, these measures converge to an explicit semi-invariant probability measure supported on the space of homothety classes of pairs of orthogonal lattices tangent to the light cone. Our main motivation is a conjecture of Sargent and Shapira, which is resolved as a special case of our general result.

Limit distributions for $\text{SO}(n,1)$ action on $k$-lattices in $\mathbb{R}^{n+1}$

TL;DR

This paper analyzes the asymptotic distribution of norm-ball averages for orbits of a lattice acting on the moduli space of pairs of orthogonal lattices in . By combining a sums-to-integrals framework, explicit volume asymptotics, and equidistribution results derived from Ratner-type theory and Margulis’ banana trick, the authors identify a limiting semi-invariant measure concentrated on degenerate pairs lying tangent to the light cone, with a precise base-fiber decomposition over degenerate Grassmannian pairs. They treat both nondegenerate and degenerate cases, including a detailed analysis of the special two-dimensional degeneracies in low dimensions, which resolves the Sargent–Shapira conjecture in a special instance. In higher dimensions, they obtain the joint limiting distribution for pairs of orthogonal lattices and provide a complete framework distinguishing commensurable versus non-commensurable scenarios, thereby advancing the understanding of lattice distributions in homogeneous dynamics. The results connect to Furstenberg-type questions on Diophantine approximation and demonstrate a robust methodology for deriving limiting measures via volume estimates and unipotent invariance, with potential applications to broader lattice counting problems in semi-simple Lie groups.

Abstract

We study the asymptotic distribution of norm ball averages along orbits of a lattice acting on the moduli space of pairs of orthogonal discrete subgroups of up to homothety. Our main result shows that, except for special -lattices in lying in hyperplanes tangent to the light cone, these measures converge to an explicit semi-invariant probability measure supported on the space of homothety classes of pairs of orthogonal lattices tangent to the light cone. Our main motivation is a conjecture of Sargent and Shapira, which is resolved as a special case of our general result.

Paper Structure

This paper contains 16 sections, 28 theorems, 259 equations.

Table of Contents

  1. Introduction
  2. Sums to integrals
  3. Volume computations
  4. Action on orthogonal lattices
  5. The case of $H_{i,j}$
  6. The case of $U$
  7. Equidistribution statements
  8. Generalities
  9. Equidsitribution of expanding pieces of $H_{i,j}$
  10. Equidistribution of unipotent pieces
  11. Case 1
  12. Proving the main results
  13. Partition of unity arguments
  14. Proof of the main results
  15. Proofs for Theorem \ref{['thm:non-degen orb']}, Theorem \ref{['thm:degen orbits higher dim']} and Theorem \ref{['thm:degen orbits low dim']} in the non-commensurable case
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $\Gamma\subset G$ be a lattice, and let $x_0:=([\Lambda_1],[\Lambda_2])\in X_{r,n+1}$ be such that $q|_{\normalfont \text{Span}_\mathbb{R}(\Lambda_1)}$ is positive definite. We denote $P_0:=\normalfont \text{Span}_{\mathbb{R}}(\Lambda_1).$ Then, for all $f\in C_c(X_{r,n+1}),$ where $d(kK_{\mathbf{P}^\infty})$ is the $K$-invariant probability on $K/K_{\mathbf{P}^\infty}$.

Theorems & Definitions (51)

  • Theorem 1.1
  • proof : Proving the conjecture of Sargent-Shapira Sargent2017DynamicsOT
  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 41 more