Symmetries of the Three Gap Theorem

TL;DR

This paper investigates the symmetries in the arrangement of gaps created by fractional parts of multiples of a real number, extending the classical Three Gap Theorem with new insights into their structural properties.

Contribution

It introduces a novel analysis of the symmetries in the sequence of gap sizes, providing deeper understanding of the geometric and combinatorial structure of the theorem.

Findings

Identifies symmetrical patterns in the order of gap sizes

Provides a new characterization of gap arrangements

Enhances understanding of the structural properties of the Three Gap Theorem

Abstract

The Three Gap Theorem states that for any $\alpha \in \mathbb{R}$ and $N \in \mathbb{N}$, the fractional parts of $\{ 0\alpha, 1\alpha, \dots, (N - 1)\alpha \}$ partition the unit circle into gaps of at most three distinct lengths. We prove a result about symmetries in the order with which the sizes of gaps appear on the circle.