  Number systems with a rational number $a/b > 1$ as base have gained interest
in recent years. In particular, relations to Mahler's 3/2-problem as well as
the Josephus problem have been established. In the present paper we show that
the patterns of digits in the representations of positive integers in such a
number system are uniformly distributed. We study the sum-of-digits function of
number systems with rational base $a/b$ and use representations w.r.t. this
base to construct normal numbers in base $a$ in the spirit of Champernowne. The
main challenge in our proofs comes from the fact that the language of the
representations of integers in these number systems is not context-free. The
intricacy of this language makes it impossible to prove our results along
classical lines. In particular, we use self-affine tiles that are defined in
certain subrings of the ad\'ele ring $\mathbb{A}_\mathbb{Q}$ and Fourier
analysis in $\mathbb{A}_\mathbb{Q}$. With help of these tools we are able to
reformulate our results as estimation problems for character sums.
