A partial orbit of a Kronecker sequence in 2 dimensions with 5 gaps.
A partial orbit of a Kronecker sequence in 2 dimensions with 5 gaps.
The two images above illustrate initial orbits of two dimensional Kronecker sequences which exhibit 5 distinct nearest neighbor distances, which is the maximum possible (hover over the images for details). The interactive model below illustrates the initial orbit of a three dimensional Kronecker sequence which exhibits 7 distinct nearest neighbor gaps. At the moment, this is the maximum number of gaps which has been discovered for such a sequence. For more details on these problems, see the paper "A five distance theorem for Kronecker sequences," which can be found at arxiv:2009.08444.

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Last update: 22 September 2020.