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ON REAL PARTS OF POWERS OF COMPLEX PISOT NUMBERS

Part of: Algebraic number theory: global fields Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  16 March 2016

TOUFIK ZAÏMI
TOUFIK ZAÏMI*
Affiliation:
Department of Mathematics and Statistics, College of Science, Al Imam Mohammad Ibn Saud Islamic University, PO Box 90950, Riyadh 11623, Saudi Arabia email tmzaemi@imamu.edu.sa
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Abstract

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We prove that a nonreal algebraic number $\unicode[STIX]{x1D703}$ with modulus greater than $1$ is a complex Pisot number if and only if there is a nonzero complex number $\unicode[STIX]{x1D706}$ such that the sequence of fractional parts $(\{\Re (\unicode[STIX]{x1D706}\unicode[STIX]{x1D703}^{n})\})_{n\in \mathbb{N}}$ has a finite number of limit points. Also, we characterise those complex Pisot numbers $\unicode[STIX]{x1D703}$ for which there is a convergent sequence of the form $(\{\Re (\unicode[STIX]{x1D706}\unicode[STIX]{x1D703}^{n})\})_{n\in \mathbb{N}}$ for some $\unicode[STIX]{x1D706}\in \mathbb{C}^{\ast }$. These results are generalisations of the corresponding real ones, due to Pisot, Vijayaraghavan and Dubickas.

Keywords

complex Pisot numbersdistribution modulo one

MSC classification

Primary: 11R06: PV-numbers and generalizations; other special algebraic numbers; Mahler measure
Secondary: 11J71: Distribution modulo one 11R80: Totally real fields
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 2 , October 2016 , pp. 245 - 253
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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