A new complexity function, repetitions in Sturmian words, and irrationality exponents of Sturmian numbers
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- by Yann Bugeaud and Dong Han Kim PDF
- Trans. Amer. Math. Soc. 371 (2019), 3281-3308 Request permission
Abstract:
We introduce and study a new complexity function in combinatorics on words, which takes into account the smallest second occurrence time of a factor of an infinite word. We characterize the eventually periodic words and the Sturmian words by means of this function. Then, we establish a new result on repetitions in Sturmian words and show that it is best possible. Let $b \ge 2$ be an integer. We deduce a lower bound for the irrationality exponent of real numbers whose sequence of $b$-ary digits is a Sturmian sequence over $\{0, 1, \ldots , b-1\}$ and we prove that this lower bound is best possible. As an application, we derive some information on the $b$-ary expansion of $\log (1 + \frac {1}{a})$ for any integer $a \ge 34$.References
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Additional Information
- Yann Bugeaud
- Affiliation: Université de Strasbourg, CNRS, IRMA, UMR 7501, 7 rue René Descartes, 67084 Strasbourg, France
- Email: bugeaud@math.unistra.fr
- Dong Han Kim
- Affiliation: Department of Mathematics Education, Dongguk University – Seoul, Seoul 04620, Korea
- MR Author ID: 630927
- Email: kim2010@dongguk.edu
- Received by editor(s): October 30, 2015
- Received by editor(s) in revised form: January 7, 2017, and August 23, 2017
- Published electronically: December 3, 2018
- Additional Notes: The second author was supported by the National Research Foundation of Korea (NRF-2015R1A2A2A01007090).
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 3281-3308
- MSC (2010): Primary 68R15; Secondary 11A63, 11J82
- DOI: https://doi.org/10.1090/tran/7378
- MathSciNet review: 3896112