Some constructions for the higher-dimensional three-distance theorem
For a given real number $\alpha$, let us place the fractional parts of the points $0, \alpha, 2 \alpha, \dots, (N-1) \alpha$ 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 $0 \leq n,m \lt N$. 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.