RATIONAL SELF-AFFINE TILES

An integral self-affine tile is the solution of a set equation AT = U-d is an element of D(T + d), where A is an n x n integer matrix and D is a finite subset of Z(n). In the recent decades, these objects and the induced tilings have been studied systematically. We extend this theory to matrices A is an element of Q(nxn). We define rational self-affine tiles as compact subsets of the open subring R-n x Pi(p) K-p of the adele ring A(K), 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 tile with a rational matrix. These tiles are defined as the intersection of a (translation of a) rational self-affine tile with Rn x Pi(p){0} similar or equal to 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, 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.