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

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