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The three gap theorem and the space of lattices. (English) Zbl 1391.11092

Summary: The three gap theorem (or Steinhaus conjecture) asserts that there are at most three distinct gap lengths in the fractional parts of the sequence \(\alpha,2\alpha,\dots,N\alpha\), for any integer \(N\) and real number \(\alpha\). This statement was proved in the 1950s independently by various authors. Here we present a different approach using the space of two-dimensional Euclidean lattices.

MSC:

11K06 General theory of distribution modulo \(1\)
11H06 Lattices and convex bodies (number-theoretic aspects)
52C05 Lattices and convex bodies in \(2\) dimensions (aspects of discrete geometry)
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References:

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