The three gap theorem and the space of lattices

Jens Marklof; Andreas Strombergsson

doi:10.4169/amer.math.monthly.124.8.741

The three gap theorem and the space of lattices

Jens Marklof, Andreas Strombergsson

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Abstract

The three gap theorem (or Steinhaus conjecture) asserts that there are at most three distinct gap lengths in the fractional parts of the sequence α, 2α, ⋯, Nα, for any integer N and real number α. 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.

Original language	English
Pages (from-to)	741-745
Number of pages	5
Journal	American Mathematical Monthly
Volume	124
Issue number	8
Early online date	1 Oct 2017
DOIs	https://doi.org/10.4169/amer.math.monthly.124.8.741
Publication status	Published - Oct 2017

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AB - The three gap theorem (or Steinhaus conjecture) asserts that there are at most three distinct gap lengths in the fractional parts of the sequence α, 2α, ⋯, Nα, for any integer N and real number α. 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.

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M3 - Article (Academic Journal)

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JO - American Mathematical Monthly

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The three gap theorem and the space of lattices

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