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Article Dans Une Revue Journal of Fourier Analysis and Applications Année : 2013

Patterns in rational base number systems

Résumé

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.
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Dates et versions

hal-00681647 , version 1 (22-03-2012)

Identifiants

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