Symmetries of power-free integers in number fields and their shift spaces

2024/07/11 by Fabian Gundlach, Gundlach, Fabian, Jürgen Klüners +1 · 1 citation
Computer Science · Mathematics · #11R04 #15A86 #37B10 #Coding theory and cryptography #Dynamical Systems (math.DS) #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2407.08438

openalex publication_date 2024/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We describe the group of \mathbb Z-linear automorphisms of the ring of integers of a number field K that preserve the set V_K,k of kth power-free integers: every such map is the composition of a field automorphism and the multiplication by a unit. We show that those maps together with translations generate the extended symmetry group of the shift space \mathbb D_K,k associated to V_K,k. Moreover, we show that no two such dynamical systems \mathbb D_K,k and \mathbb D_L,l are topologically conjugate and no one is a factor system of another. We generalize the concept of kth power-free integers to sieves and study the resulting admissible shift spaces.

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