A two dimensional analog of the three gap theorem

TL;DR

This paper explores the geometric structure of Voronoi cells formed by points generated from two irrational, independent numbers on a 2D torus, extending the classical three gap theorem to two dimensions.

Contribution

It introduces a two-dimensional analog of the three gap theorem, analyzing Voronoi cell areas and shapes for points derived from irrational vectors.

Findings

Voronoi cell areas exhibit specific distribution patterns

Cell shapes vary depending on the irrational vector components

The study provides experimental insights into 2D gap phenomena

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.