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Pair correlations of logarithms of complex lattice points

Jouni Parkkonen, Frédéric Paulin

TL;DR

This work develops a comprehensive framework for pair correlations of complex logarithms of lattice points under varying scalings and multiplicities. By embedding the problem in an abelian group setting and using the complex log map, the authors establish existence and characterize phase transitions: Poissonian behaviour for sublinear scalings, total mass loss for superlinear scalings, and a distinct level-repulsion regime at linear scaling. Introducing Euler weights ties the arithmetic of imaginary quadratic fields to geometric questions in hyperbolic 3-manifolds, notably linking weighted logarithmic correlations to the distribution of lengths of common perpendiculars between cusp neighborhoods in Bianchi orbifolds. The paper yields explicit limiting densities and rigorous error controls across multiple regimes, with strong geometric consequences and uniform convergence results on compact subsets of grid spaces. These results illuminate the interplay between number theory, discrete geometry, and hyperbolic geometry, providing a precise probabilistic description of fine-scale spacings in logarithmic lattice settings.

Abstract

We study the correlations of pairs of complex logarithms of $\mathbb Z$-lattice points in the complex line at various scalings, proving the existence of pair correlation functions. We prove that at the linear scaling, the pair correlations exhibit level repulsion, as it sometimes occurs in statistical physics. We prove total loss of mass phenomena at superlinear scalings, and Poissonian behaviour at sublinear scalings. The case of Euler weights has applications to the pair correlation of the lengths of common perpendicular geodesic arcs from the maximal Margulis cusp neighborhood to itself in the Bianchi orbifold $\mathrm{PSL}_2(\mathbb Z[i]) \backslash\mathbb H^3_{\mathbb R}$.

Pair correlations of logarithms of complex lattice points

TL;DR

This work develops a comprehensive framework for pair correlations of complex logarithms of lattice points under varying scalings and multiplicities. By embedding the problem in an abelian group setting and using the complex log map, the authors establish existence and characterize phase transitions: Poissonian behaviour for sublinear scalings, total mass loss for superlinear scalings, and a distinct level-repulsion regime at linear scaling. Introducing Euler weights ties the arithmetic of imaginary quadratic fields to geometric questions in hyperbolic 3-manifolds, notably linking weighted logarithmic correlations to the distribution of lengths of common perpendiculars between cusp neighborhoods in Bianchi orbifolds. The paper yields explicit limiting densities and rigorous error controls across multiple regimes, with strong geometric consequences and uniform convergence results on compact subsets of grid spaces. These results illuminate the interplay between number theory, discrete geometry, and hyperbolic geometry, providing a precise probabilistic description of fine-scale spacings in logarithmic lattice settings.

Abstract

We study the correlations of pairs of complex logarithms of -lattice points in the complex line at various scalings, proving the existence of pair correlation functions. We prove that at the linear scaling, the pair correlations exhibit level repulsion, as it sometimes occurs in statistical physics. We prove total loss of mass phenomena at superlinear scalings, and Poissonian behaviour at sublinear scalings. The case of Euler weights has applications to the pair correlation of the lengths of common perpendicular geodesic arcs from the maximal Margulis cusp neighborhood to itself in the Bianchi orbifold .
Paper Structure (7 sections, 22 theorems, 178 equations)

This paper contains 7 sections, 22 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. Pair correlation of grid points without weight or scaling
  3. Pair correlation of grid points with scaling without weight
  4. Mertens and Mirsky formulae for algebraic number fields
  5. Pair correlation of integral lattice points with Euler weight and no scaling
  6. Pair correlation of integral lattice points with scaling and Euler weight
  7. Pair correlations of common perpendiculars in the Bianchi manifolds $\operatorname{PSL}({\cal O}_K)\backslash{{\mathbb H}}^3_{\mathbb R}$

Key Result

Theorem 1.1

Let $\alpha\ge 0$ and let $\Lambda$ be a ${\mathbb Z}$-grid. As $N\rightarrow+\infty$, the normalized pair correlation measures $\frac{1}{N^{4-2\alpha}}\; {\cal R}_N^{\,{\cal L}_\Lambda,\,\operatorname{id}^\alpha}$ on the cylinder $E_N={\mathbb C}/(2\pi i N^\alpha {\mathbb Z})$ converge for the poin The convergence is uniform on every compact subset of ${\mathbb Z}$-grids $\Lambda$ for the Chabaut

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Theorem 2.2
  • Remark 2.3
  • Corollary 2.4
  • Corollary 2.5
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • ...and 13 more