Rational self-affine tiles associated to standard and nonstandard digit systems

TL;DR

This paper investigates rational self-affine tiles generated by digit systems with expanding matrices, providing criteria for positive measure, analyzing topological properties, and establishing tiling theorems in a very general setting.

Contribution

It introduces a general framework for rational self-affine tiles associated with both standard and nonstandard digit systems, extending previous results to broader classes.

Findings

Criteria for positive Haar measure of self-affine sets

Topological properties and tiling theorems for these sets

Extension to nonstandard digit systems and rational matrices

Abstract

We consider digit systems $(A,\mathcal{D})$, where $ A \in \mathbb{Q}^{n\times n}$ is an expanding matrix and the digit set $\mathcal{D}$ is a suitable subset of $\mathbb{Q}^n$. To such a system, we associate a self-affine set $\mathcal{F} = \mathcal{F}(A,\mathcal{D})$ that lives in a certain representation space $\mathbb{K}$. If $A$ is an integer matrix, then $\mathbb{K} = \mathbb{R}^n$, while in the general rational case $\mathbb{K}$ contains an additional solenoidal factor. We give a criterion for $\mathcal{F}$ to have positive Haar measure, i.e., for being a rational self-affine tile. We study topological properties of $\mathcal{F}$ and prove some tiling theorems. Our setting is very general in the sense that we allow $(A,\mathcal{D})$ to be a nonstandard digit system. A standard digit system $(A,\mathcal{D})$ is one in which we require $\mathcal{D}$ 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.