Some constructions for the higher-dimensional three-distance theorem

TL;DR

This paper extends the three-distance theorem to two dimensions, analyzing the distribution of points on the circle formed by linear combinations of two real numbers, and provides examples with finitely many and infinitely many interval lengths.

Contribution

It introduces a two-dimensional version of the three-distance theorem and constructs examples with both finitely and infinitely many interval lengths.

Findings

Finitely many lengths, up to seven, can occur for certain pairs of real numbers.

Examples include pairs where the numbers are not badly approximable.

Infinite lengths can also occur in the two-dimensional setting.

Abstract

For a given real number $\alpha$, let us place the fractional parts of the points $0, \alpha, 2 \alpha,$ $ \cdots, (N-1) \alpha$ on the unit circle. These points partition the unit circle into intervals having at most three lengths, one being the sum of the other two. This is the three distance theorem. We consider a two-dimensional version of the three distance theorem obtained by placing on the unit circle the points $ n\alpha+ m\beta $, for $0 \leq n,m < N$. We provide examples of pairs of real numbers $(\alpha,\beta)$, with $1,\alpha, \beta$ rationally independent, for which there are finitely many lengths between successive points (and in fact, seven lengths), with $(\alpha,\beta)$ not badly approximable, as well as examples for which there are infinitely many lengths.