Patterns in Rational Base Number Systems

Number systems with a rational number a / b >1 as base have gained interest in recent years. In particular, relations to Mahler’s -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 adèle ring and Fourier analysis in . With help of these tools we are able to reformulate our results as estimation problems for character sums.