| Title:
|
On the limit points of the fractional parts of powers of Pisot numbers (English) |
| Author:
|
Dubickas, Artūras |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
42 |
| Issue:
|
2 |
| Year:
|
2006 |
| Pages:
|
151-158 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Pisot numbers |
| Keyword:
|
fractional parts |
| Keyword:
|
limit points |
| MSC:
|
11J71 |
| MSC:
|
11R06 |
| idZBL:
|
Zbl 1164.11026 |
| idMR:
|
MR2240352 |
| . |
| Date available:
|
2008-06-06T22:47:45Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107991 |
| . |
| Reference:
|
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| Reference:
|
[2] Cassels J. W. S.: An introduction to Diophantine approximation.Cambridge University Press, 1957. Zbl 0077.04801, MR 0087708 |
| Reference:
|
[3] Dubickas A.: A note on powers of Pisot numbers.Publ. Math. Debrecen 56 (2000), 141–144. Zbl 0999.11035, MR 1740499 |
| Reference:
|
[4] Dubickas A.: Integer parts of powers of Pisot and Salem numbers.Arch. Math. (Basel) 79 (2002), 252–257. Zbl 1004.11059, MR 1944949 |
| Reference:
|
[5] Dubickas A.: Sequences with infinitely many composite numbers.Analytic and Probabilistic Methods in Number Theory, Palanga 2001 (eds. A. Dubickas, A. Laurinčikas and E. Manstavičius), TEV, Vilnius (2002), 57–60. Zbl 1049.11072, MR 1964849 |
| Reference:
|
[6] Dubickas A.: Arithmetical properties of powers of algebraic numbers.Bull. London Math. Soc. 38 (2006), 70–80. Zbl 1164.11025, MR 2201605 |
| Reference:
|
[7] Flatto L., Lagarias J. C., Pollington A. D.: On the range of fractional parts $\lbrace \xi (p/q)^n\rbrace $.Acta Arith. 70 (1995), 125–147. MR 1322557 |
| Reference:
|
[8] Kuba G.: The number of lattice points below a logarithmic curve.Arch. Math. (Basel) 69 (1997), 156–163. Zbl 0899.11050, MR 1458702 |
| Reference:
|
[9] Luca F.: On a question of G. Kuba.Arch. Math. (Basel) 74 (2000), 269–275. Zbl 0995.11043, MR 1742638 |
| Reference:
|
[10] Smyth C. J.: The conjugates of algebraic integers.Amer. Math. Monthly 82 (1975), 86. |
| Reference:
|
[11] Zaimi T.: An arithmetical property of powers of Salem numbers.J. Number Theory (to appear). Zbl 1147.11037, MR 2256803 |
| . |
