## An arithmetical property of powers of Salem numbers.(English)Zbl 1147.11037

In [Bull. Lond. Math. Soc. 38, No. 1, 70–80 (2006; Zbl 1164.11025)] the reviewer established a lower bound for the difference $$\Delta(\zeta,\alpha)$$ between the largest and the smallest limit points of the sequence of fractional parts $$\{\zeta \alpha^n\},$$ $$n=0,1,2,\dots,$$ where $$\alpha>1$$ is an algebraic number and $$\zeta \neq 0.$$ An additional requirement is that $$\zeta \notin {\mathbb Q}(\alpha)$$ in case $$\alpha$$ is a Salem or a Pisot number.
Here, the author considers one of these exceptional cases, namely, the case when $$\alpha$$ is a Salem number (say, of length $$L(\alpha)$$) and $$\zeta$$ is a nonzero element of the field $${\mathbb Q}(\alpha)$$. He proves that $$\Delta(\zeta,\alpha) \geq 1/L(\alpha)$$ if $$\alpha-1$$ is a unit. If, alternatively, $$\alpha-1$$ is not a unit then he shows that for every $$\varepsilon>0$$ there is a $$\zeta \in {\mathbb Q}(\alpha)$$ such that $$\Delta(\zeta,\alpha)<\varepsilon$$. Furthermore, he proves that $$\Delta(\zeta,\alpha)=1$$ if $$\zeta$$ is an algebraic integer, so that both $$0$$ and $$1$$ are the limit points of the sequence $$\{\zeta \alpha^n\},$$ $$n=0,1,2,\dots.$$ For $$\zeta=1,$$ an old result of Salem implies that this sequence is everywhere dense in $$[0,1]$$. In his proof the author uses arguments from linear algebra, diophantine analysis and uniform distribution.

### MSC:

 11J71 Distribution modulo one 11R04 Algebraic numbers; rings of algebraic integers 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure

### Keywords:

Salem number; fractional parts

Zbl 1164.11025
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### References:

 [1] A. Dubickas, Arithmetical properties of powers of algebraic numbers, Bull. London Math. Soc., submitted for publication · Zbl 1164.11025 [2] A. Dubickas, On the limits of the fractional parts of powers of Pisot numbers, Arch. Math. (Brno), submitted for publication · Zbl 1164.11026 [3] Luca, F., On a question of G. kuba, Arch. math., 74, 269-275, (2000) · Zbl 0995.11043 [4] Salem, R., Algebraic numbers and Fourier analysis, Heath math. monographs, (1963), Heath Boston · Zbl 0126.07802 [5] Serre, J.P., Local fields, (1979), Springer Berlin [6] Zaïmi, T., Remarks on certain salem numbers, Arab. J. math. sci., 7, 1, 1-10, (2001) · Zbl 0987.11062
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