Regular Structures in Kronecker Permutations
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918135592910848 |
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| author | Clément, François |
| author_facet | Clément, François |
| contents | Kronecker sequences $(k α\mod 1)_{k=1}^{\infty}$ for some irrational $α> 0$ have played an important role in many areas of mathematics. It is possible to associate to each finite segment $(k α\mod 1)_{k=1}^{n}$ a permutation $π\in S_n$ associated with the canonical lifting to two dimensions. We show that these permutations induced by Kronecker sequences based on irrational $α$ are extremely regular for specific choices of $n$ and $α$. In particular, all quadratic irrationals have an infinite number of choices of $n$ that lead to permutations where no cycle has length more than 4. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_03782 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regular Structures in Kronecker Permutations Clément, François Combinatorics Number Theory Kronecker sequences $(k α\mod 1)_{k=1}^{\infty}$ for some irrational $α> 0$ have played an important role in many areas of mathematics. It is possible to associate to each finite segment $(k α\mod 1)_{k=1}^{n}$ a permutation $π\in S_n$ associated with the canonical lifting to two dimensions. We show that these permutations induced by Kronecker sequences based on irrational $α$ are extremely regular for specific choices of $n$ and $α$. In particular, all quadratic irrationals have an infinite number of choices of $n$ that lead to permutations where no cycle has length more than 4. |
| title | Regular Structures in Kronecker Permutations |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2509.03782 |