|Date||Sep 12, 2017|
For a given real number $\alpha$, let us place the fractional parts of the points $0, \alpha, 2 \alpha, \cdots, N \alpha$ on the unit circle.
These points partition the unit circle into $N+1$ 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 the points $ n\alpha+ m\beta $, for $0 \leq n,m \leq N$, on the unit circle. We provide examples of pairs of real numbers $(\alpha,\beta)$ for which there are finitely many lengths between successive points (with $1,\alpha, \beta$ rationally independent and not badly approximable), as well as examples for which there are infinitely many of them.