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Search: MSC category 11J71 ( Distribution modulo one [See also 11K06] )

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1. CJM 2007 (vol 59 pp. 503)

Chevallier, Nicolas
Cyclic Groups and the Three Distance Theorem
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 $q$.

Categories:11J70, 11J71, 11J13

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