Rational self-affine tiles

Abstract

An integral self-affine tile is the solution of a set equation A T = ⋃ d ∈ D ( T + d ) \mathbf {A} \mathcal {T} = \bigcup _{d \in \mathcal {D}} (\mathcal {T} + d) , where A \mathbf {A} is an n × n n \times n integer matrix and D \mathcal {D} is a finite subset of Z n \mathbb {Z}^n . In the recent decades, these objects and the induced tilings have been studied systematically. We extend this theory to matrices A ∈ Q n × n \mathbf {A} \in \mathbb {Q}^{n \times n} . We define rational self-affine tiles as compact subsets of the open subring R n × ∏ p K p \mathbb {R}^n\times \prod _\mathfrak {p} K_\mathfrak {p} of the adèle ring A K \mathbb {A}_K , where the factors of the (finite) product are certain p \mathfrak {p} -adic completions of a number field K K that is defined in terms of the characteristic polynomial of A \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 R n × ∏ p { 0 } ≃ R n \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 D \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.