# Farey Sequences for Thin Groups

## TL;DR

This paper introduces generalized Farey sequences linked to thin groups, demonstrating their equidistribution, gap distribution convergence, and applications in Diophantine approximation, with explicit formulas and ergodic properties for specific cases.

## Contribution

It develops the concept of generalized Farey sequences arising from thin groups and analyzes their distribution, gap statistics, and ergodic properties, extending classical number theory results.

## Key findings

- Sequences equidistribute and gap distribution converges.
- Explicit formula for gap distribution in a specific example.
- Constructs an ergodic measure analogous to the Gauss measure.

## Abstract

The classical Farey sequence of height $Q$ is the set of rational numbers in reduced form with denominator less than $Q$. In this paper we introduce the concept of a generalized Farey sequence. While these sequences arise naturally in the study of discrete (and in particular thin) subgroups, they can be used to study interesting number theoretic sequences - for example rationals whose continued fraction partial quotients are subject to congruence conditions. We show that these sequences equidistribute, that the gap distribution converges, and we answer an associated problem in Diophantine approximation with Fuchsian groups. Moreover, for one specific example, we use a sparse Ford configuration construction to write down an explicit formula for the gap distribution. Finally for this example, we construct the analogue of the Gauss measure in this context which we show is ergodic for the Gauss map. This allows us to prove a theorem about the Gauss-Kuzmin statistics of the sequence.