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Poissonian pair correlation for directions in multi-dimensional affine lattices, and escape of mass estimates for embedded horospheres

Wooyeon Kim, Jens Marklof

Abstract

We prove the convergence of moments of the number of directions of affine lattice vectors that fall into a small disc, under natural Diophantine conditions on the shift. Furthermore, we show that the pair correlation function is Poissonian for any irrational shift in dimension 3 and higher, including well-approximable vectors. Convergence in distribution was already proved in the work of Strömbergsson and the second author, and the principal step in the extension to convergence of moments is an escape of mass estimate for averages over embedded $\operatorname{SL}(d,\mathbb{R})$-horospheres in the space of affine lattices.

Poissonian pair correlation for directions in multi-dimensional affine lattices, and escape of mass estimates for embedded horospheres

Abstract

We prove the convergence of moments of the number of directions of affine lattice vectors that fall into a small disc, under natural Diophantine conditions on the shift. Furthermore, we show that the pair correlation function is Poissonian for any irrational shift in dimension 3 and higher, including well-approximable vectors. Convergence in distribution was already proved in the work of Strömbergsson and the second author, and the principal step in the extension to convergence of moments is an escape of mass estimate for averages over embedded -horospheres in the space of affine lattices.
Paper Structure (11 sections, 20 theorems, 161 equations)

This paper contains 11 sections, 20 theorems, 161 equations.

Table of Contents

  1. Introduction
  2. Limit distribution and higher moments
  3. The space of affine lattices
  4. Properties of the limiting distribution
  5. Escape of mass
  6. The main lemma
  7. Proof of Corollary \ref{['cor1']} and Corollary \ref{['cor2']}
  8. Proof of Corollary \ref{['cor1']}
  9. Proof of Corollary \ref{['cor2']}
  10. Brjuno type condition
  11. Counterexamples

Key Result

Theorem 1.1

Let $d\geq 2$ and $\boldsymbol{\xi}\in\mathbb{R}^d\setminus\mathbb{Q}^d$; furthermore if $d=2$ assume that $\boldsymbol{\xi}$ is $(0,0,2)$-vaguely Diophantine. Then the pair correlation function of the sequence $(\boldsymbol{\upsilon}_j)_{j=1}^\infty$ of directions is Poissonian.

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Corollary 2.5
  • Lemma 3.1
  • proof
  • Proposition 5.1
  • Lemma 5.2
  • ...and 23 more