Symmetries of the Three-Gap Theorem

Abstract

The Three-Gap Theorem states that for any 𝛼∈ℝ and 𝑁∈ℕ, the fractional parts of {0⁢𝛼,1⁢𝛼,…,(𝑁−1)⁢𝛼} partition the unit circle into gaps of at most three distinct lengths. It is also of interest to find patterns in how the order of different gap sizes appear as one goes counterclockwise around the circle. This note is devoted to proving a result about symmetries in this ordering.