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Cyclic Groups and the Three Distance Theorem

  Published:2007-06-01
 Printed: Jun 2007
  • Nicolas Chevallier
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Abstract

We give a two dimensional extension of the three distance Theorem. Let $\theta$ be in $\mathbf{R}^{2}$ and let $q$ be in $\mathbf{N}$. There exists a triangulation of $\mathbf{R}^{2}$ invariant by $\mathbf{Z}^{2}$-translations, whose set of vertices is $\mathbf{Z}^{2}+\{0,\theta,\dots,q\theta\}$, and whose number of different triangles, up to translations, is bounded above by a constant which does not depend on $\theta$ and $q$.
MSC Classifications: 11J70, 11J71, 11J13 show english descriptions Continued fractions and generalizations [See also 11A55, 11K50]
Distribution modulo one [See also 11K06]
Simultaneous homogeneous approximation, linear forms
11J70 - Continued fractions and generalizations [See also 11A55, 11K50]
11J71 - Distribution modulo one [See also 11K06]
11J13 - Simultaneous homogeneous approximation, linear forms
 

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