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On the distance from a rational power to the nearest integer. (English) Zbl 1097.11035

Not many things are known about the distribution modulo 1 of the sequence \(\{\xi(3/2)^n\}\) for given values of \(\xi\). In the nice paper under review the author proves that the interval [0.23, 0.77] contains at least one limit joint of the sequence \(\{\xi(3/2)^n\}\) for any nonzero real number \(\xi\). He also proves that the distance to the nearest integer of \(\xi(p/q)^n\), for any rational \(p/q>1\), has both “small” and “large” limit points. Similar results hold for \(\xi\alpha^m\), where \(\alpha >1\) is an algebraic number, and \(\xi\neq 0\) is a real number that does not belong to \(\mathbb{Q}(\alpha)\) if \(\alpha\) is a Pisot or a Salem number. Curiously enough the Thue-Morse sequence enters the picture.
Please note that three references appeared: [1] see S. D. Adhikari et al., Acta Arith. 119, No. 4, 307–316 (2005: Zbl 05001723),[14] A. Dubickas [Bull. Lond. Math. Soc. 38, No. 1, 70–80 (2006; Zbl 1164.11025)], [17] A. Dubickas and A. Novikas Math. Z. 251, No. 3, 635–648 (2005; Zbl 1084.11009)].
Please also note that Reference [4] could be replaced (or completed) by a much more ancient paper of the same authors: J.-P. Allouche and M. Cosnard, C. R. Acad. Sci. Paris. Sér. I. 296, 159–162 (1983; Zbl 0547.58027)], see also “Théorie des nombres et automates”, Thèse d’État, Bordeaux (1983) by the reviewer).

MSC:

11J71 Distribution modulo one
11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11B85 Automata sequences
11K06 General theory of distribution modulo \(1\)
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References:

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