  Recently, generalizations of the classical Three Gap Theorem to higher
dimensions attracted a lot of attention. In particular, upper bounds for the
number of nearest neighbor distances have been established for the Euclidean
and the maximum metric. It was proved that a generic multi-dimensional
Kronecker attains the maximal possible number of different gap lengths for
every sub-exponential subsequence. We mirror this result in dimension $d \in
\left\{ 2, 3 \right\}$ by constructing Kronecker sequences which have a
surprisingly low number of different nearest neighbor distances for infinitely
$N \in \mathbb{N}$. Our proof relies on simple arguments from the theory of
continued fractions.
