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Rational self-affine tiles

Authors: Wolfgang Steiner and Jörg M. Thuswaldner
Journal: Trans. Amer. Math. Soc. 367 (2015), 7863-7894
MSC (2010): Primary 52C22, 11A63, 28A80
Published electronically: March 13, 2015
MathSciNet review: 3391902
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Abstract: An integral self-affine tile is the solution of a set equation $ \mathbf {A} \mathcal {T} = \bigcup _{d \in \mathcal {D}} (\mathcal {T} + d)$, where $ \mathbf {A}$ is an $ n \times n$ integer matrix and $ \mathcal {D}$ is a finite subset of $ \mathbb{Z}^n$. In the recent decades, these objects and the induced tilings have been studied systematically. We extend this theory to matrices $ \mathbf {A} \in \mathbb{Q}^{n \times n}$. We define rational self-affine tiles as compact subsets of the open subring $ \mathbb{R}^n\times \prod _\mathfrak{p} K_\mathfrak{p}$ of the adèle ring $ \mathbb{A}_K$, where the factors of the (finite) product are certain $ \mathfrak{p}$-adic completions of a number field $ K$ that is defined in terms of the characteristic polynomial of  $ \mathbf {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 tile with a rational matrix. These tiles are defined as the intersection of a (translation of a) rational self-affine tile with $ \mathbb{R}^n \times \prod _\mathfrak{p} \{0\} \simeq \mathbb{R}^n$. 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 the digit set $ \mathcal {D}$, intersection tiles are instances of tiles defined in terms of shift radix systems and canonical number systems. This enables us to gain new results for tilings associated with numeration systems.

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Additional Information

Wolfgang Steiner
Affiliation: LIAFA, CNRS UMR 7089, Université Paris Diderot – Paris 7, Case 7014, 75205 Paris Cedex 13, France

Jörg M. Thuswaldner
Affiliation: Chair of Mathematics and Statistics, University of Leoben, A-8700 Leoben, Austria

Keywords: Self-affine tile, tiling, shift radix system
Received by editor(s): February 19, 2012
Received by editor(s) in revised form: August 14, 2013
Published electronically: March 13, 2015
Additional Notes: This research was supported by the Austrian Science Foundation (FWF), project S9610, which is part of the national research network FWF–S96 “Analytic combinatorics and probabilistic number theory”, and by the Amadée grant FR 16/2010 – PHC Amadeus 2011 “From fractals to numeration”.
Dedicated: Dedicated to Professor Shigeki Akiyama on the occasion of his $50^{th}$ birthday
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.