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

Abstract

The Three Gap Theorem states that for any $\alpha \in (0,1)$ and any integer $N\ge 1$, the fractional parts of the sequence $0,\alpha ,2\alpha ,\cdots ,(N-1)\alpha$ partition the unit interval into $N$ subintervals having at most \emph{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.

arXiv commentReorganization, a new section on generating primitive lattice points, and a proof of the explicit continuous distribution result. This is work in progress

References in corpus (1)

• Ergodic Properties of Compositions of Interval Exchange Maps and Rotations