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arXiv · 1708.04380

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

Diaaeldin Taha
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Abstract

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 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.

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Diaaeldin Taha. 2017-08-15. The Three Gap Theorem, Interval Exchange Transformations, and Zippered Rectangles. https://arxiv.org/abs/1708.04380

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