Gaps in the complex Farey sequence of an imaginary quadratic number field

TL;DR

The paper addresses the asymptotic gap distribution for complex Farey fractions in imaginary quadratic fields by translating the problem into a homogeneous-dynamics setting and leveraging a joint equidistribution result in $\mathbb H^3_{\mathbb R}$. The main contribution is proving the existence of a density $\mu$ on $[1,\infty)$ describing the gap statistics, with an explicit integral formula for the cumulative distribution and a sharp tail decay of order $\delta^{-4}$, verified in Gaussian and Eisenstein cases. The method combines a dynamical reformulation of gaps via a cone intersection problem with a lifted joint equidistribution for complex Farey fractions, yielding a rigorous link between arithmetic gaps and flows on homogenous spaces. The results illuminate the fine-scale structure of complex Farey sequences and connect to classical gap laws (e.g., Hall-type densities), with potential extensions to other fields, ideal constraints, and higher-dimensional Farey-type systems. Overall, the work advances the understanding of deterministic gap statistics in higher-dimensional modular settings through homogeneous dynamics techniques.

Abstract

Given an imaginary quadratic number field $K$ with ring of integers $\mathcal{O}_K$, we are interested in the asymptotic \emph{distance to nearest neighbour} (or \emph{gap}) statistic of complex Farey fractions $\frac{p}{q}$, with $p,q \in \mathcal{O}_K$ and $0<|q|\leq T$, as $T \to \infty$. Reformulating this problem in a homogeneous dynamical setting, we follow the approach of J. Marklof for real Farey fractions with several variables (2013) and adapt a joint equidistribution result in the real $3$-dimensional hyperbolic space of J. Parkkonen and F. Paulin (2023) to derive the existence of a probability measure describing this asymptotic gap statistic. We obtain an integral formula for the associated cumulative distribution function, and use geometric arguments to find an explicit estimate for its tail distribution in the cases of Gaussian and Eisenstein fractions.

Figures (6)

• Figure 1: Gaussian fractions with height at most $10$ (on the left) and $20$ (on the right) in the square torus ${\mathbb C}/{\mathbb Z}[i]$, and colours of points depending on their height.
• Figure 2: A numerical approximation of the asymptotic gap density for Gaussian fractions, using points with height at most $50$. For the approached tail distribution function, see the top graph in Figure \ref{['fig:tail_approx_zi']}.
• Figure 3: Points of ${\cal F}_t$ with $K={\mathbb Q}(i\sqrt{3})$ (Eisenstein fractions) hence ${\cal O}_K={\mathbb Z}[e^{i\frac{2\pi}{3}}]$, with height at most $20$ and colours of points depending on their height.
• Figure 4: An example of a decomposition $\{{\cal X}_1, {\cal X}_2\}$ of $({\mathbb C}/{\mathbb Z}[i]) \smallsetminus {\cal X}$ (resp. $\{{\cal X}_1,{\cal X}_2,{\cal X}_3\}$ of $({\mathbb C}/{\mathbb Z}[j]) \smallsetminus {\cal X}$) into parts which are each homeomorphic to $({\mathbb Z}[i]' \backslash {\mathbb C})\smallsetminus \operatorname{pr}_\bullet({\cal X})$ (resp. to $({\mathbb Z}[j]' \backslash {\mathbb C})\smallsetminus \operatorname{pr}_\bullet({\cal X})$). The sets ${\cal X}$ and $\operatorname{pr}_\bullet({\cal X})$ are represented by black lines and black dots.
• Figure 5: At the top, the empirical tail distribution $\delta \to \mu_t(\, ]\delta, +\infty[\,)$ (in blue) and the graph of $\delta \mapsto \frac{1}{\delta^4}$ (in red) using Gaussian fractions and the height value $e^\frac{t}{2}=30$. At the bottom, a logarithmic version of the top graph: we illustrate the estimate \ref{['eq:tail_estimate_zij']} by comparing the functions $\ell \mapsto \ln(\mu_t(]e^\ell, +\infty[\,))$ and $\ell \mapsto -4\ell$.

Theorems & Definitions (17)

• Theorem 1.1
• Definition 2.1
• Theorem 2.2
• Corollary 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Theorem 2.6