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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 - Continued fractions and generalizations [See also 11A55, 11K50] 11J71 - Distribution modulo one [See also 11K06] 11J13 - Simultaneous homogeneous approximation, linear forms