Patterns in Rational Base Number Systems |
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Authors: | Johannes F. Morgenbesser Wolfgang Steiner Jörg M. Thuswaldner |
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Affiliation: | 1. Fakult?t für Mathematik, Universit?t Wien, Nordbergstrasse 15, 1090, Wien, Austria 2. LIAFA, CNRS UMR 7089, Université Paris Diderot—Paris 7, Case 7014, 75205, Paris Cedex 13, France 3. Department of Mathematics, 275 TMCB, Brigham Young University, Provo, UT, 84602, USA
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Abstract: | 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. |
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