Symmetries 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.

To appear in The American Math Monthly. This is the author's original manuscript (AOM), and this version does not include several improvements that took place due to the refereeing process. Comments welcome!

References in corpus (1)

• Nearest Neighbor Distances on a Circle: Multidimensional Case