Geodesic
Multi-dimensional Kronecker sequences with a small number of gap lengths
Diskretnaya Matematika, Tome 33 (2021) no. 4, pp. 11-18

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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.} \communicated{
Keywords: Kronecker Sequences, Nearest Neighbor Distance, Continued Fractions. 
@article{DM_2021_33_4_a1,
     author = {Ch. Weiss},
     title = {Multi-dimensional {Kronecker} sequences with a small number of gap lengths},
     journal = {Diskretnaya Matematika},
     pages = {11--18},
     publisher = {mathdoc},
     volume = {33},
     number = {4},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2021_33_4_a1/}
}
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Ch. Weiss. Multi-dimensional Kronecker sequences with a small number of gap lengths. Diskretnaya Matematika, Tome 33 (2021) no. 4, pp. 11-18. http://geodesic.mathdoc.fr/item/DM_2021_33_4_a1/