Regular Structures in Kronecker Permutations

2025/09/04 by Clément, François
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2509.03782

Abstract

Kronecker sequences (k α\mod 1)_k=1^∞ 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=1ⁿ a permutation π∈ 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.

Citations

• Searching permutations for constructing uniformly distributed point sets
• On the Global Optimality of Fibonacci Lattices in the Torus
• Constructing Optimal L_∞ Star Discrepancy Sets
• A five distance theorem for Kronecker sequences
• The three gap theorem and the space of lattices
• Optimal point sets for quasi-Monte Carlo integration of bivariate periodic functions with bounded mixed derivatives
• On the Small Ball Inequality in All Dimensions

Related

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• The lattice permutation condition for Kronecker tableaux (Extended Abstract)
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