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The Three Gap Theorem states that for any $\alpha \in (0,1)$ and any integer$N \geq 1$, 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 theoremusing zippered rectangles, and present a new gaps theorem (along with twoproofs) for sequences generated as orbits of general interval exchangetransformations. We also derive a number of results on primitive points inlattices mirroring several properties of Farey fractions. This makes itpossible to derive a previously known, explicit distribution result related tothe Three Gap Theorem using ergodic theory.

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