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Euclidean algorithms are Gaussian over imaginary quadratic fields

Dohyeong Kim, Jungwon Lee, Seonhee Lim

TL;DR

This paper extends Baladi–Vallée Gaussian limit theorems for division steps and costs of the Euclidean algorithm from real rationals to imaginary quadratic fields by analyzing the complex Hurwitz continued fraction map. It develops a robust dynamical framework built around a finite Markov partition and a transfer-operator acting on a carefully constructed piecewise $C^1$ space, establishing a spectral gap and Dolgopyat-type estimates to deduce Gaussian limit laws. The authors prove central limit theorems for both continuous trajectories and $K$-rational trajectories, along with residual equidistribution modulo $q$, by connecting moment generating functions to Dirichlet-series and applying Tauberian arguments. The work advances the probabilistic understanding of Euclidean algorithms in algebraic number fields and provides tools relevant to complex Diophantine approximation and modular-symbol-related phenomena.

Abstract

The distributional analysis of Euclidean algorithms was carried out by Baladi and Vallée. They showed the asymptotic normality of the number of division steps and associated costs in the Euclidean algorithm as a random variable on the set of rational numbers with bounded denominator based on the transfer operator methods. We extend their result to the Euclidean algorithm over appropriate imaginary quadratic fields by studying dynamics of the nearest integer complex continued fraction map, which is piecewise analytic and expanding but not a full branch map. By observing a finite Markov partition with a regular CW-structure, which enables us to associate the transfer operator acting on a direct sum of spaces of $C^1$-functions, we obtain the limit Gaussian distribution as well as residual equidistribution.

Euclidean algorithms are Gaussian over imaginary quadratic fields

TL;DR

This paper extends Baladi–Vallée Gaussian limit theorems for division steps and costs of the Euclidean algorithm from real rationals to imaginary quadratic fields by analyzing the complex Hurwitz continued fraction map. It develops a robust dynamical framework built around a finite Markov partition and a transfer-operator acting on a carefully constructed piecewise space, establishing a spectral gap and Dolgopyat-type estimates to deduce Gaussian limit laws. The authors prove central limit theorems for both continuous trajectories and -rational trajectories, along with residual equidistribution modulo , by connecting moment generating functions to Dirichlet-series and applying Tauberian arguments. The work advances the probabilistic understanding of Euclidean algorithms in algebraic number fields and provides tools relevant to complex Diophantine approximation and modular-symbol-related phenomena.

Abstract

The distributional analysis of Euclidean algorithms was carried out by Baladi and Vallée. They showed the asymptotic normality of the number of division steps and associated costs in the Euclidean algorithm as a random variable on the set of rational numbers with bounded denominator based on the transfer operator methods. We extend their result to the Euclidean algorithm over appropriate imaginary quadratic fields by studying dynamics of the nearest integer complex continued fraction map, which is piecewise analytic and expanding but not a full branch map. By observing a finite Markov partition with a regular CW-structure, which enables us to associate the transfer operator acting on a direct sum of spaces of -functions, we obtain the limit Gaussian distribution as well as residual equidistribution.
Paper Structure (26 sections, 42 theorems, 180 equations, 4 figures, 1 table)

This paper contains 26 sections, 42 theorems, 180 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Complex continued fraction maps and its transfer operator
  4. Complex Gauss dynamical system
  5. Metric properties of inverse branch
  6. Finite Markov partition with cell structure
  7. Cell structures on $I$ and function spaces
  8. Cell structure from lines and circles
  9. Compatibility of $\mathcal{P}$ and $T$
  10. Natural extension and dual inverse branch
  11. Spectral gap of the transfer operators on piecewise $C^1$-space
  12. Function space: Norms on $C^1(\mathcal{P})$
  13. Sufficient conditions for quasi-compactness
  14. Ruelle--Perron--Frobenius Theorem
  15. A priori bounds for the normalised family
  16. ...and 11 more sections

Key Result

Theorem 1.1

For a digit cost $c$ with a moderate growth $c(a)=O(\log a)$, the distribution of the total cost$C$ on the set is asymptotically Gaussian, with the speed of convergence $O(1/\sqrt{\log N})$ as $N \to \infty.$

Figures (4)

  • Figure 1: For $d=3$, the domain $I$ is the hexagon in the center. The inversion $z \mapsto 1/z$ maps $I$ to the region outside the green circles. The grey hexagon $I+1$ lies in the union of the interior of circles, thus $O_1$ is empty. The green hexagon $I+ \frac{3+\sqrt{-3}}{2}$ intersects with some circles, thus $O_{\frac{3+\sqrt{-3}}{2}} \neq I.$
  • Figure 2: Examples of 0, 1, 2-cells in a finite partition ${{\mathcal{P}}}$ depicted in green ($d=1$).
  • Figure 3: Partition element $O_{1+i}$ and image $TO_{1+i}$ (as a disjoint union of cells in a finite partition ${{\mathcal{P}}}$) depicted in grey ($d=1$).
  • Figure 4: Boundary of cells in ${{\mathcal{P}}}$ induced by a circle in $W_0 \cup W_1$ intersecting $I+(1+2i)$, and all images of $h_{1+2i}(P)$ inside $O_{1+2i}$ depicted in grey $(d=1)$.

Theorems & Definitions (86)

  • Theorem 1.1: Baladi--Vallée bal:val
  • Theorem 1: Theorem \ref{['clt:cont']}
  • Theorem 2: Theorem \ref{['clt:discrete']}
  • Theorem 3: Theorem \ref{['thm:equid']}
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Remark 1.6
  • Theorem 4: Theorem \ref{['thm:ruelle']} and \ref{['main:dolgopyat']}
  • ...and 76 more