The Three Gap Theorem and Riemannian Geometry

TL;DR

This paper extends the classical Three Gap Theorem from circle rotations to isometries on compact Riemannian manifolds, exploring the distribution of points along geodesics and revealing new geometric insights.

Contribution

It generalizes the Three Gap Theorem to Riemannian manifolds, connecting circle rotation properties to manifold isometries and geodesic distributions.

Findings

At most three distinct distances between consecutive points on geodesics

Generalization of the Three Gap Theorem to Riemannian settings

New geometric bounds for point distributions along geodesics

Abstract

The classical Three Gap Theorem asserts that for a natural number n and a real number p, there are at most three distinct distances between consecutive elements in the subset of [0,1) consisting of the reductions modulo 1 of the first n multiples of p. Regarding it as a statement about rotations of the circle, we find results in a similar spirit pertaining to isometries of compact Riemannian manifolds and the distribution of points along their geodesics.