The Three Gap Theorem and Riemannian geometry

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.