# On the limit points of the fractional parts of powers of Pisot numbers

Archivum Mathematicum (2006)

• Volume: 042, Issue: 2, page 151-158
• ISSN: 0044-8753

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## Abstract

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We consider the sequence of fractional parts $\left\{\xi {\alpha }^{n}\right\}$, $n=1,2,3,\cdots$, where $\alpha >1$ is a Pisot number and $\xi \in ℚ\left(\alpha \right)$ is a positive number. We find the set of limit points of this sequence and describe all cases when it has a unique limit point. The case, where $\xi =1$ and the unique limit point is zero, was earlier described by the author and Luca, independently.

## How to cite

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Dubickas, Artūras. "On the limit points of the fractional parts of powers of Pisot numbers." Archivum Mathematicum 042.2 (2006): 151-158. <http://eudml.org/doc/249821>.

@article{Dubickas2006,
abstract = {We consider the sequence of fractional parts $\lbrace \xi \alpha ^n\rbrace$, $n=1,2,3,\dots$, where $\alpha >1$ is a Pisot number and $\xi \in \{\mathbb \{Q\}\}(\alpha )$ is a positive number. We find the set of limit points of this sequence and describe all cases when it has a unique limit point. The case, where $\xi =1$ and the unique limit point is zero, was earlier described by the author and Luca, independently.},
author = {Dubickas, Artūras},
journal = {Archivum Mathematicum},
keywords = {Pisot numbers; fractional parts; limit points; Pisot numbers; fractional parts; limit points},
language = {eng},
number = {2},
pages = {151-158},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On the limit points of the fractional parts of powers of Pisot numbers},
url = {http://eudml.org/doc/249821},
volume = {042},
year = {2006},
}

TY - JOUR
AU - Dubickas, Artūras
TI - On the limit points of the fractional parts of powers of Pisot numbers
JO - Archivum Mathematicum
PY - 2006
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 042
IS - 2
SP - 151
EP - 158
AB - We consider the sequence of fractional parts $\lbrace \xi \alpha ^n\rbrace$, $n=1,2,3,\dots$, where $\alpha >1$ is a Pisot number and $\xi \in {\mathbb {Q}}(\alpha )$ is a positive number. We find the set of limit points of this sequence and describe all cases when it has a unique limit point. The case, where $\xi =1$ and the unique limit point is zero, was earlier described by the author and Luca, independently.
LA - eng
KW - Pisot numbers; fractional parts; limit points; Pisot numbers; fractional parts; limit points
UR - http://eudml.org/doc/249821
ER -

## References

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7. Flatto L., Lagarias J. C., Pollington A. D., On the range of fractional parts $\left\{\xi {\left(p/q\right)}^{n}\right\}$, Acta Arith. 70 (1995), 125–147. (1995) MR1322557
8. Kuba G., The number of lattice points below a logarithmic curve, Arch. Math. (Basel) 69 (1997), 156–163. (1997) Zbl0899.11050MR1458702
9. Luca F., On a question of G. Kuba, Arch. Math. (Basel) 74 (2000), 269–275. Zbl0995.11043MR1742638
10. Smyth C. J., The conjugates of algebraic integers, Amer. Math. Monthly 82 (1975), 86. (1975)
11. Zaimi T., An arithmetical property of powers of Salem numbers, J. Number Theory (to appear). Zbl1147.11037MR2256803

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