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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

Citations:

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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