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

TL;DR

This paper investigates the asymptotic distribution of gaps between complex Farey fractions in imaginary quadratic fields, extending dynamical and geometric methods to derive explicit formulas and tail estimates.

Contribution

It adapts a joint equidistribution approach to complex Farey fractions in imaginary quadratic fields, providing explicit formulas and tail estimates for the gap distribution.

Findings

Derived an integral formula for the gap distribution function.

Established the existence of a probability measure for the asymptotic gap statistic.

Provided explicit tail estimates for Gaussian and Eisenstein fractions.

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.