Multi-dimensional Kronecker sequences with a small number of gap lengths
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Christian Weiß
Abstract
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 ∈ {2, 3} by constructing Kronecker sequences which have a surprisingly low number of different nearest neighbor distances for infinitely N ∈ ℕ. Our proof relies on simple arguments from the theory of continued fractions.
Note: Originally published in Diskretnaya Matematika (2021) 33,№4, 11–18 (in Russian).
Acknowledgments
Parts of the research on this paper was conducted during a stay of the author at the Max-Planck Institute in Bonn whom he would like to thank for hospitality and an inspiring scientific atmosphere. Moreover, I would like to thank the referee for his careful reading and valuable comments.
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Articles in the same Issue
- Frontmatter
- Single diagnostic tests for inversion faults of gates in circuits over arbitrary bases
- Generalized de Bruijn graphs
- A family of asymptotically independent statistics in polynomial scheme containing the Pearson statistic
- Linear recurrent relations, power series distributions, and generalized allocation scheme
- Variance of the number of cycles of random A-permutation
- Multi-dimensional Kronecker sequences with a small number of gap lengths
Articles in the same Issue
- Frontmatter
- Single diagnostic tests for inversion faults of gates in circuits over arbitrary bases
- Generalized de Bruijn graphs
- A family of asymptotically independent statistics in polynomial scheme containing the Pearson statistic
- Linear recurrent relations, power series distributions, and generalized allocation scheme
- Variance of the number of cycles of random A-permutation
- Multi-dimensional Kronecker sequences with a small number of gap lengths
