| Title:
|
Comments on the fractional parts of Pisot numbers (English) |
| Author:
|
Zaïmi, Toufik |
| Author:
|
Selatnia, Mounia |
| Author:
|
Zekraoui, Hanifa |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
51 |
| Issue:
|
3 |
| Year:
|
2015 |
| Pages:
|
153-161 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Pisot numbers |
| Keyword:
|
fractional parts |
| Keyword:
|
limit points |
| MSC:
|
11J71 |
| MSC:
|
11R04 |
| MSC:
|
11R06 |
| idZBL:
|
Zbl 06487027 |
| idMR:
|
MR3397268 |
| DOI:
|
10.5817/AM2015-3-153 |
| . |
| Date available:
|
2015-09-09T09:44:53Z |
| Last updated:
|
2016-04-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144426 |
| . |
| Reference:
|
[1] Bertin, M.J., Decomps-Guilloux, A., Grandet-Hugo, M., Pathiaux-Delefosse, M., Schreiber, J.P.: Pisot and Salem numbers.Birkhäuser Verlag Basel, 1992. MR 1187044 |
| Reference:
|
[2] Boyd, D.W.: Linear recurrence relations for some generalized Pisot sequences.Advances in Number Theory, Proc. of the 1991 CNTA Conference, Oxford University Press, 1993, pp. 333–340. Zbl 0790.11012, MR 1368431 |
| Reference:
|
[3] Bugeaud, Y.: An introduction to diophantine approximation.Cambridge University Press, Cambridge, 2012. MR 2953186 |
| Reference:
|
[4] Dubickas, A.: On the limit points of the fractional parts of powers of Pisot numbers.Arch. Math. (Brno) 42 (2006), 151–158. Zbl 1164.11026, MR 2240352 |
| Reference:
|
[5] Marcus, D.: Number fields.Springer, Berlin, 1977, 3rd edition. Zbl 0383.12001, MR 0457396 |
| Reference:
|
[6] Smyth, C.J.: The conjugates of algebraic integers.Amer. Math. Monthly 82 (86) (1975). |
| Reference:
|
[7] Zaïmi, T.: Comments on the distribution modulo one of powers of Pisot and Salem numbers.Publ. Math. Debrecen 80 (2012), 417–426. Zbl 1263.11067, MR 2943014, 10.5486/PMD.2012.5098 |
| Reference:
|
[8] Zaïmi, T.: On the spectra of Pisot numbers.Glasgow Math. J. 54 (2012), 127–132. Zbl 1303.11118, MR 2862390, 10.1017/S0017089511000462 |
| . |
