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Cyclic Groups and the Three Distance Theorem

Published online by Cambridge University Press:  20 November 2018

Nicolas Chevallier
Nicolas Chevallier*
Affiliation:
Université de Haute Alsace, 4, rue des frères Lumière, 68093 Mulhouse, France email: [email protected]
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Abstract

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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 ${{\mathbf{Z}}^{2}}\,+\,\{0,\,\theta ,\,\ldots ,\,q\theta \}$, and whose number of different triangles, up to translations, is bounded above by a constant which does not depend on $\theta $ and $q$.

Keywords

11J7011J7111J13
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 3 , 01 June 2007 , pp. 503 - 552
Copyright
Copyright © Canadian Mathematical Society 2007

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