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Fractional parts of powers and Sturmian words. (English. Abridged French version) Zbl 1140.11318

From the text: Let \(b\geq 2\) be an integer. In terms of combinatorics on words we describe all irrational numbers \(\xi > 0\) with the property that the fractional parts \(\{\xi b^n\}\), \(n\geq 0\), all belong to a semi-open or an open interval of length \(1/b\). The length of such an interval cannot be smaller, that is, for irrational \(\xi\), the fractional parts \(\{\xi b^n\}\), \(n\geq 0\), cannot all belong to an interval of length smaller than \(1/b\).
Moreover all these numbers lie in an interval of length \(1/b\) if and only if the (transcendental) number \(\xi\) is equal to \(g+k/(b-1)+t_b(w)\) where \(g\) is an arbitrary integer, \(k\in\{0,1,\dots,b-2\}\), \(w\) is a Sturmian word over \(\{0,1\}\), and \(t_b(w)=\sum_{j=1}^{\infty}w_jb^{-j}\).

MSC:

11B85 Automata sequences
11K06 General theory of distribution modulo \(1\)
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
05A05 Permutations, words, matrices
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References:

[1] S. Akiyama, Ch. Frougny, J. Sakarovitch, On number representation in a rational base, submitted for publication; S. Akiyama, Ch. Frougny, J. Sakarovitch, On number representation in a rational base, submitted for publication
[2] Berstel, J.; Karhumäki, J., Combinatorics on words – a tutorial, Bull. EATCS, 79, 178-228 (2003) · Zbl 1169.68560
[3] Bugeaud, Y., Linear mod one transformations and the distribution of fractional parts \(\{\xi(p / q)^n \}\), Acta Arith., 114, 301-311 (2004) · Zbl 1061.11041
[4] A. Dubickas, Arithmetical properties of powers of algebraic numbers, Bull. London Math. Soc., in press; A. Dubickas, Arithmetical properties of powers of algebraic numbers, Bull. London Math. Soc., in press · Zbl 1164.11025
[5] A. Dubickas, On the distance from a rational power to the nearest integer, submitted for publication; A. Dubickas, On the distance from a rational power to the nearest integer, submitted for publication · Zbl 1097.11035
[6] A. Dubickas, Arithmetical properties of linear recurrent sequences, submitted for publication; A. Dubickas, Arithmetical properties of linear recurrent sequences, submitted for publication · Zbl 1111.11044
[7] A. Dubickas, A. Novikas, Integer parts of powers of rational numbers, Math. Z., in press; A. Dubickas, A. Novikas, Integer parts of powers of rational numbers, Math. Z., in press · Zbl 1084.11009
[8] Ferenczi, S.; Mauduit, Ch., Transcendence of numbers with a low complexity expansion, J. Number Theory, 67, 146-161 (1997) · Zbl 0895.11029
[9] Flatto, L.; Lagarias, J. C.; Pollington, A. D., On the range of fractional parts \(\{\xi(p / q)^n \}\), Acta Arith., 70, 125-147 (1995) · Zbl 0821.11038
[10] Lothaire, M., Algebraic Combinatorics on Words, Encyclopedia Math. Appl., vol. 90 (2002), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1001.68093
[11] Mahler, K., An unsolved problem on the powers of \(3 / 2\), J. Austral. Math. Soc., 8, 313-321 (1968) · Zbl 0155.09501
[12] Morse, M.; Hedlund, G. A., Symbolic dynamics II: Sturmian sequences, Amer. J. Math., 62, 1-42 (1940) · JFM 66.0188.03
[13] A. Schinzel, On the reduced length of a polynomial with real coefficients, submitted for publication; A. Schinzel, On the reduced length of a polynomial with real coefficients, submitted for publication
[14] T. Zaimi, An arithmetical property of powers of Salem numbers, submitted for publication; T. Zaimi, An arithmetical property of powers of Salem numbers, submitted for publication · Zbl 1147.11037
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