Cyclic Groups and the Three Distance Theorem

http://dx.doi.org/10.4153/CJM-2007-022-3
Canad. J. Math. 59(2007), 503-552

  Published:2007-06-01
 Printed: Jun 2007

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 $q$.