Geodesic
Comments on the fractional parts of Pisot numbers
Archivum mathematicum, Tome 51 (2015) no. 3, pp. 153-161 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.5817/AM2015-3-153
Classification : 11J71, 11R04, 11R06
Keywords: Pisot numbers; fractional parts; limit points 
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     title = {Comments on the fractional parts of {Pisot} numbers},
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Zaïmi, Toufik; Selatnia, Mounia; Zekraoui, Hanifa. Comments on the fractional parts of Pisot numbers. Archivum mathematicum, Tome 51 (2015) no. 3, pp. 153-161. doi: 10.5817/AM2015-3-153

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