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Complex continued fractions, Kleinian and extremal theory for cusp excursions

Alexander Baumgartner, Mark Pollicott

TL;DR

The paper extends extreme-value theory for real continued fractions to complex continued fractions with digits in the Euclidean rings $\mathfrak{o}_d$ ($d\in\{1,2,3,7,11\}$), proving a Fréchet law for the maximal digit. It then interprets this digit-extremal behavior through the geometry of cusp excursions on 3D Bianchi orbifolds, establishing a Gumbel/Fréchet-type limit for excursion heights via a detailed excursion-timing analysis and invariant-measure methods. By connecting complex continued fractions to geodesic flows on hyperbolic 3-space, the work delivers explicit constants $\kappa_d$ that tie cusp excursion statistics to hyperbolic-volume data, and derives Sullivan–Nakada-type Diophantine approximation results in imaginary quadratic fields with precise limiting distributions. The resulting framework provides a coherent bridge between complex continued fraction dynamics, hyperbolic dynamics, and metric Diophantine approximation in imaginary quadratic settings, with explicit formulas for constants and measurable limit laws.

Abstract

For the each of the five Euclidean rings of complex quadratic integers, we consider a complex continued fraction algorithm with digits in the ring. We show for each algorithm that the maximal digit obeys a Fréchet distribution. We use this to find a limiting distribution for cusp excursions on Bianchi orbifolds associated with the aforementioned rings of quadratic integers.

Complex continued fractions, Kleinian and extremal theory for cusp excursions

TL;DR

The paper extends extreme-value theory for real continued fractions to complex continued fractions with digits in the Euclidean rings (), proving a Fréchet law for the maximal digit. It then interprets this digit-extremal behavior through the geometry of cusp excursions on 3D Bianchi orbifolds, establishing a Gumbel/Fréchet-type limit for excursion heights via a detailed excursion-timing analysis and invariant-measure methods. By connecting complex continued fractions to geodesic flows on hyperbolic 3-space, the work delivers explicit constants that tie cusp excursion statistics to hyperbolic-volume data, and derives Sullivan–Nakada-type Diophantine approximation results in imaginary quadratic fields with precise limiting distributions. The resulting framework provides a coherent bridge between complex continued fraction dynamics, hyperbolic dynamics, and metric Diophantine approximation in imaginary quadratic settings, with explicit formulas for constants and measurable limit laws.

Abstract

For the each of the five Euclidean rings of complex quadratic integers, we consider a complex continued fraction algorithm with digits in the ring. We show for each algorithm that the maximal digit obeys a Fréchet distribution. We use this to find a limiting distribution for cusp excursions on Bianchi orbifolds associated with the aforementioned rings of quadratic integers.
Paper Structure (12 sections, 29 theorems, 135 equations, 3 figures)

This paper contains 12 sections, 29 theorems, 135 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Complex continued fractions
  3. Extremal value theory for the digits of continued fractions
  4. Bianchi orbifolds
  5. From a Fréchet law to cusp excursions
  6. Excursion times
  7. Proof of Theorem \ref{['theorem: cusp excursions on Euclidean Bianchi Orbifolds']}
  8. From cusp excursions to Diophantine approximation
  9. The Liouville measure on $\mathrm{T}^1\mathbf{H}^3$
  10. Calculating the constant in Theorem \ref{['theorem: Gumbel Law for Bianchi Groups']}.
  11. Diophantine approximation
  12. Intersection Times and Cusp Excursions

Key Result

Proposition 1.1

For regular continued fractions, if $u>0$, then where $d\nu = (\log2)^{-1}dz/(1+z)$ is the Gauss measure on $[0,1]$.

Figures (3)

  • Figure 1: The partition $\bigcup_{k=1}^J V_k$ for $d=3$.
  • Figure 2: Construction of the fundamental domain for $d=2$. The points $x+yi+rj\in \mathcal{F}_2$ belong to the region bounded by the opaque hemisphere and the four vertical rectangles.
  • Figure 3: The geodesic $\tilde{\gamma}_n$ on $\mathbf{H}^3$ which arcs from the point $z_1^{(n)}$ to $z_2^{(n)}$. It intersects the spheres $H(a_n)$ and $H(0)$ at time $t_{n-1}$ and $t_{n}$ respectively.

Theorems & Definitions (54)

  • Proposition 1.1: Galambos
  • Proposition 1.2: Kirsebom
  • Theorem 1.1: Fréchet Law for the Maximal Digit
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Proposition 1.3: Pollicott
  • Proposition 1.4: Sullivan
  • Theorem 1.3
  • Definition 2.1
  • ...and 44 more