Comments on the fractional parts of Pisot numbers
Toufik Zaïmi; Mounia Selatnia; Hanifa Zekraoui
Archivum Mathematicum (2015)
- Volume: 051, Issue: 3, page 153-161
- ISSN: 0044-8753
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topZaïmi, Toufik, Selatnia, Mounia, and Zekraoui, Hanifa. "Comments on the fractional parts of Pisot numbers." Archivum Mathematicum 051.3 (2015): 153-161. <http://eudml.org/doc/271634>.
@article{Zaïmi2015,
abstract = {Let $L(\theta ,\lambda )$ be the set of limit points of the fractional parts $\lbrace \lambda \theta ^\{n\}\rbrace $, $n=0,1,2, \dots $, where $\theta $ is a Pisot number and $\lambda \in \mathbb \{Q\}(\theta )$. Using a description of $L(\theta ,\lambda )$, due to Dubickas, we show that there is a sequence $(\lambda _\{n\})_\{n\ge 0\}$ of elements of $\mathbb \{Q\}(\theta )$ such that $\operatorname\{Card\}\,(L(\theta ,\lambda _\{n\}))< \operatorname\{Card\}\,(L(\theta ,\lambda _\{n+1\}))$, $\forall $$n\ge 0$. Also, we prove that the fractional parts of Pisot numbers, with a fixed degree greater than 1, are dense in the unit interval.},
author = {Zaïmi, Toufik, Selatnia, Mounia, Zekraoui, Hanifa},
journal = {Archivum Mathematicum},
keywords = {Pisot numbers; fractional parts; limit points},
language = {eng},
number = {3},
pages = {153-161},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Comments on the fractional parts of Pisot numbers},
url = {http://eudml.org/doc/271634},
volume = {051},
year = {2015},
}
TY - JOUR
AU - Zaïmi, Toufik
AU - Selatnia, Mounia
AU - Zekraoui, Hanifa
TI - Comments on the fractional parts of Pisot numbers
JO - Archivum Mathematicum
PY - 2015
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 051
IS - 3
SP - 153
EP - 161
AB - Let $L(\theta ,\lambda )$ be the set of limit points of the fractional parts $\lbrace \lambda \theta ^{n}\rbrace $, $n=0,1,2, \dots $, where $\theta $ is a Pisot number and $\lambda \in \mathbb {Q}(\theta )$. Using a description of $L(\theta ,\lambda )$, due to Dubickas, we show that there is a sequence $(\lambda _{n})_{n\ge 0}$ of elements of $\mathbb {Q}(\theta )$ such that $\operatorname{Card}\,(L(\theta ,\lambda _{n}))< \operatorname{Card}\,(L(\theta ,\lambda _{n+1}))$, $\forall $$n\ge 0$. Also, we prove that the fractional parts of Pisot numbers, with a fixed degree greater than 1, are dense in the unit interval.
LA - eng
KW - Pisot numbers; fractional parts; limit points
UR - http://eudml.org/doc/271634
ER -
References
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- Zaïmi, T., 10.1017/S0017089511000462, Glasgow Math. J. 54 (2012), 127–132. (2012) Zbl1303.11118MR2862390DOI10.1017/S0017089511000462
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