Cyclic Groups and the Three Distance Theorem
Nicolas Chevallier · Canadian Journal of Mathematics · 2007
Abstract We give a two dimensional extension of the three distance theorem. Let θ be in R2 and let q be in N. There exists a triangulation of R2 invariant by Z2-translations, whose set of vertices is Z2 + ﹛0, θ, … , qθ﹜, and whose number of different triangles, up to translations, is bounded above by a constant which does not depend on θ and q.