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