  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.

Number Theory;Dynamical Systems