Journal of Fourier Analysis and Applications

, Volume 19, Issue 2, pp 225–250 | Cite as

Patterns in Rational Base Number Systems

  • Johannes F. Morgenbesser
  • Wolfgang Steiner
  • Jörg M. Thuswaldner


Number systems with a rational number a/b>1 as base have gained interest in recent years. In particular, relations to Mahler’s \(\frac{3}{2}\)-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 adèle 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.


Rational number system Normal numbers p-adic numbers Fourier analysis, Sum-of-digits function 

Mathematics Subject Classification (2010)

11A63 28A80 52C22 


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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Johannes F. Morgenbesser
    • 1
  • Wolfgang Steiner
    • 2
  • Jörg M. Thuswaldner
    • 3
  1. 1.Fakultät für MathematikUniversität WienWienAustria
  2. 2.LIAFA, CNRS UMR 7089Université Paris Diderot—Paris 7Paris Cedex 13France
  3. 3.Department of Mathematics, 275 TMCBBrigham Young UniversityProvoUSA

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