Address:
Corresponding author: T. Zaïmi, Al-Imam Muhammad Ibn Saud Islamic University, College of Science, Department of Mathematics and Statistics, P.O. Box 90950, Riyadh 11623, Saudi Arabia,
M. Selatnia, H. Zekraoui, Larbi Ben M’hidi University, Faculty of Science, Department of Mathematics and Informatic, Oum El Bouaghi 04000, Algeria
E-mail:
tmzaemi@imamu.edu.sa
toufikzaimi@yahoo.com
mouniaselatnia@yahoo.fr
hzekraoui421@gmail.com
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.
AMSclassification: primary 11J71; secondary 11R06, 11R04.
Keywords: Pisot numbers, fractional parts, limit points.
DOI: 10.5817/AM2015-3-153