## 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 $$1$$

### Citations:

Zbl 1084.11009; Zbl 0547.58027; Zbl 05001723; Zbl 1164.11025
Full Text:

### References:

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