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ON THE EXPANSIONS OF REAL NUMBERS IN TWO MULTIPLICATIVELY DEPENDENT BASES

Part of: Discrete mathematics in relation to computer science Elementary number theory

Published online by Cambridge University Press:  01 December 2016

YANN BUGEAUD  and
DONG HAN KIM
YANN BUGEAUD
Affiliation:
IRMA, U.M.R. 7501, Université de Strasbourg et CNRS, 7 rue René Descartes, 67084 Strasbourg, France email bugeaud@math.unistra.fr
DONG HAN KIM*
Affiliation:
Department of Mathematics Education, Dongguk University–Seoul, 30 Pildong-ro 1-gil, Jung-gu, Seoul 04620, Korea email kim2010@dongguk.edu
*
✉kim2010@dongguk.edu
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Abstract

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Let $r\geq 2$ and $s\geq 2$ be multiplicatively dependent integers. We establish a lower bound for the sum of the block complexities of the $r$-ary expansion and the $s$-ary expansion of an irrational real number, viewed as infinite words on $\{0,1,\ldots ,r-1\}$ and $\{0,1,\ldots ,s-1\}$, and we show that this bound is best possible.

Keywords

combinatorics on wordsSturmian wordcomplexityb-ary expansion

MSC classification

Primary: 11A63: Radix representation; digital problems
Secondary: 68R15: Combinatorics on words
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 3 , June 2017 , pp. 373 - 383
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

Footnotes

This work was supported by the National Research Foundation of Korea (NRF-2015R1A2A2A01007090) and the research program of Dongguk University, 2016.

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