# Patterns in rational base number systems

Abstract : 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 adéle 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.
Keywords :
Document type :
Journal articles

https://hal.archives-ouvertes.fr/hal-00681647
Contributor : Wolfgang Steiner Connect in order to contact the contributor
Submitted on : Thursday, March 22, 2012 - 3:40:59 AM
Last modification on : Saturday, November 20, 2021 - 3:49:30 AM
Long-term archiving on: : Saturday, June 23, 2012 - 2:23:00 AM

### Files

rational_patterns.pdf
Files produced by the author(s)

### Citation

Johannes Morgenbesser, Wolfgang Steiner, Jörg Thuswaldner. Patterns in rational base number systems. Journal of Fourier Analysis and Applications, Springer Verlag, 2013, 19 (2), pp.225-250. ⟨10.1007/s00041-012-9246-1⟩. ⟨hal-00681647⟩

Record views