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On the spectra of Pisot numbers. (English) Zbl 1303.11118

Summary: Let \(\theta \) be a real number greater than 1, and let \(((\cdot))\) be the fractional part function. Then, \(\theta \) is said to be a \(Z\)-number if there is a non-zero real number \(\lambda \) such that \(((\lambda\theta^n)) < \tfrac 12\) for all \(n\in\mathbb N\). A. Dubickas [Glasg. Math. J. 48, No. 2, 331–336 (2006; Zbl 1138.11026)] showed that strong Pisot numbers are \(Z\)-numbers. Here it is proved that \(\theta\) is a strong Pisot number if and only if there exists \(\lambda\neq 0\) such that \(((\lambda \alpha )) < \tfrac 12\) for all \(\alpha\in\left\{\theta^n\mid n\in \mathbb{N}\right\} \cup \left\{\sum_{n=0}^N\theta^n\mid N\in\mathbb{N}\right\}\). Also, the following characterisation of Pisot numbers among real numbers greater than 1 is shown: \(\theta \) is a Pisot number \(\Leftrightarrow\exists\lambda \neq 0\) such that \(\| \lambda \alpha \| <\tfrac 13\) for all \(\alpha\in\left\{\sum_{n=0}^N a_n\theta^n\mid a_n\in \{0,1\}, N\in \mathbb N\right\}\), where \(\| \lambda \alpha \| = \min\{((\lambda \alpha)), 1- ((\lambda \alpha ))\}\).

MSC:

11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
11J71 Distribution modulo one

Citations:

Zbl 1138.11026
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References:

[1] Boyd, Advances in Number Theory, Proceedings of the 1991 CNTA Conference pp 333– (1993)
[2] DOI: 10.1007/978-3-0348-8632-1 · doi:10.1007/978-3-0348-8632-1
[3] DOI: 10.1017/S0017089506003090 · Zbl 1138.11026 · doi:10.1017/S0017089506003090
[4] Salem, Algebraic numbers and Fourier analysis (1963) · Zbl 0126.07802
[5] Pisot, Annali Scuola Norm. Sup. Pisa. 7 pp 205– (1938)
[6] DOI: 10.2307/2319148 · doi:10.2307/2319148
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