A two dimensional analog of the three gap theorem

Abstract

For a two dimensional vector $\bar v=(\alpha,\beta)$, where $\alpha>0, \beta>0$ are irrational numbers independent over $\mathbb{Q}$, we consider the set $D_n=\{(i\alpha\,{\rm mod}\,1,i\beta\,{\rm mod}\,1),i=1,\ldots,n\}$ in a two dimensional torus and the partition of this torus into Voronoi cells. Areas and forms of these cells are the subject of this experimental work.