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

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.