Regular Structures in Kronecker Permutations

Kronecker sequences $(k\alpha \mathrm{m}\mathrm{o}\mathrm{d}1{)}_{k=1}^{\mathrm{\infty }}$ for some irrational $\alpha >0$ have played an important role in many areas of mathematics. It is possible to associate to each finite segment $(k\alpha \mathrm{m}\mathrm{o}\mathrm{d}1{)}_{k=1}^n$ a permutation $\pi \in S_n$ associated with the canonical lifting to two dimensions. We show that these permutations induced by Kronecker sequences based on irrational $\alpha$ are extremely regular for specific choices of $n$ and $\alpha$. 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.