The Three Gap Theorem, Interval Exchange Transformations, and Zippered Rectangles

The Three Gap Theorem states that for any α∈ (0,1) and any integer N ≥ 1, the fractional parts of the sequence 0, α, 2α, ⋯, (N-1)α partition the unit interval into N subintervals having at most three distinct lengths. We here provide a new proof of this theorem using zippered rectangles, and present a new gaps theorem (along with two proofs) for sequences generated as orbits of general interval exchange transformations. We also derive a number of results on primitive points in lattices mirroring several properties of Farey fractions. This makes it possible to derive a previously known, explicit distribution result related to the Three Gap Theorem using ergodic theory.