2012/03/04 by Wolfgang Steiner, Steiner, Wolfgang, Jörg Μ. Thuswaldner +2 · 2 citations
Computer Science · Mathematics · #Cellular Automata and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.DS #math.NT #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1203.0758
openalex publication_date 2012/03/04 · arxiv created 2013/08/30 · arxiv updated 2013/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
An integral self-affine tile is the solution of a set equation A T = \bigcupd ∈ D (T + d), where A is an n × n integer matrix and D is a finite subset of ℤn. In the recent decades, these objects and the induced tilings have been studied systematically. We extend this theory to matrices A ∈ ℚn × n. We define rational self-affine tiles as compact subsets of the open subring ℝn× ∏_\mathfrakp K_\mathfrakp of the adéle ring \mathbbAK, where the factors of the (finite) product are certain \mathfrakp-adic completions of a number field K that is defined in terms of the characteristic polynomial of A. Employing methods from classical algebraic number theory, Fourier analysis in number fields, and results on zero sets of transfer operators, we establish a general tiling theorem for these tiles. We also associate a second kind of tiles with a rational matrix. These tiles are defined as the intersection of a (translation of a) rational self-affine tile with ℝn × ∏_\mathfrakp \0\ ≃ ℝn. Although these intersection tiles have a complicated structure and are no longer self-affine, we are able to prove a tiling theorem for these tiles as well. For particular choices of digit sets, intersection tiles are instances of tiles defined in terms of shift radix systems and canonical number systems. Therefore, we gain new results for tilings associated with numeration systems.