An integral self-affine tile is the solution of a set equation AT=⋃d∈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×∏pKp of the ad\'ele ring AK, where the factors of the
(finite) product are certain p-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×∏p{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