Fine-scale statistics for the multidimensional Farey sequence

TL;DR

This paper extends the understanding of fine-scale statistics, like gap distributions, from one-dimensional Farey sequences to higher dimensions using advanced lattice and horosphere equidistribution techniques.

Contribution

It introduces a generalized framework for analyzing multidimensional Farey sequences, building on and extending previous one-dimensional results and methods.

Findings

Generalization of gap distribution results to multiple dimensions

Use of horosphere equidistribution in lattice spaces

Framework applicable to various dimensions

Abstract

We generalize classical results on the gap distribution (and other fine-scale statistics) for the one-dimensional Farey sequence to arbitrary dimension. This is achieved by exploiting the equidistribution of horospheres in the space of lattices, and the equidistribution of Farey points in a certain subspace of the space of lattices. The argument follows closely the general approach developed by A. Str\"ombergsson and the author [Annals of Math. 172 (2010) 1949--2033].