Deducing Three Gap Theorem From Rauzy-Veech Induction

TL;DR

This paper demonstrates how the Three Gap Theorem, which limits the number of distinct gap lengths among points on a circle, can be derived using Rauzy-Veech induction, providing a new proof approach.

Contribution

It introduces a novel proof of the Three Gap Theorem based on Rauzy-Veech induction, connecting dynamical systems techniques to classical number theory.

Findings

The Three Gap Theorem can be derived from Rauzy-Veech induction.

The proof simplifies understanding of the gap structure on a circle.

New connections between interval exchange transformations and classical theorems.

Abstract

The Three Gap Theorem states that there are at most three distinct lengths of gaps if one places $n$ points on a circle, at angles of $z, 2z, 3z, \ldots nz$ from the starting point. The theorem was first proven in 1958 by S\'os and many proofs have been found since then. In this note we show how the Three Gap Theorem can easily be deduced by using Rauzy-Veech induction.