2021/10/18 by Rossi, Lucía, Wolfgang Steiner, Steiner, Wolfgang +1 · 2 citations
Computer Science · Mathematics · #11A63 #FOS: Mathematics #Mathematical Analysis and Transform Methods #Mathematical Dynamics and Fractals #Number Theory (math.NT) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2110.09112
openalex publication_date 2021/10/18 · openalex created_date 2021/10/25 · openalex updated_date 2026/07/28
We consider digit systems (A,D), where A ∈ ℚn× n is an expanding matrix and the digit set D is a suitable subset of ℚn. To such a system, we associate a self-affine set F = F(A,D) that lives in a certain representation space \mathbbK. If A is an integer matrix, then \mathbbK = ℝn, while in the general rational case \mathbbK contains an additional solenoidal factor. We give a criterion for F to have positive Haar measure, i.e., for being a rational self-affine tile. We study topological properties of F and prove some tiling theorems. Our setting is very general in the sense that we allow (A,D) to be a nonstandard digit system. A standard digit system (A,D) is one in which we require D to be a complete system of residue class representatives w.r.t. a certain naturally chosen residue class ring. Our tools comprise the Frobenius normal form and character theory of locally compact abelian groups.