Skip to Main content Skip to Navigation
Journal articles

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.
Document type :
Journal articles
Complete list of metadata

https://hal.archives-ouvertes.fr/hal-00681647
Contributor : Wolfgang Steiner <>
Submitted on : Thursday, March 22, 2012 - 3:40:59 AM
Last modification on : Wednesday, November 4, 2020 - 10:22:05 PM
Long-term archiving on: : Saturday, June 23, 2012 - 2:23:00 AM

Files

rational_patterns.pdf
Files produced by the author(s)

Identifiers

Collections

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⟩

Share

Metrics

Record views

758

Files downloads

778