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

Artūras Dubickas

Archivum Mathematicum (2006)

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

Abstract

top
We consider the sequence of fractional parts { ξ α n } , n = 1 , 2 , 3 , ⋯ , where α > 1 is a Pisot number and ξ ∈ ℚ ( α ) 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 ξ = 1 and the unique limit point is zero, was earlier described by the author and Luca, independently.

How to cite

top

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

top
  1. Bugeaud Y., Linear mod one transformations and the distribution of fractional parts { ξ ( p / q ) n } , Acta Arith. 114 (2004), 301–311. MR2101819
  2. Cassels J. W. S., An introduction to Diophantine approximation, Cambridge University Press, 1957. (1957) Zbl0077.04801MR0087708
  3. Dubickas A., A note on powers of Pisot numbers, Publ. Math. Debrecen 56 (2000), 141–144. Zbl0999.11035MR1740499
  4. Dubickas A., Integer parts of powers of Pisot and Salem numbers, Arch. Math. (Basel) 79 (2002), 252–257. Zbl1004.11059MR1944949
  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. Zbl1049.11072MR1964849
  6. Dubickas A., Arithmetical properties of powers of algebraic numbers, Bull. London Math. Soc. 38 (2006), 70–80. Zbl1164.11025MR2201605
  7. Flatto L., Lagarias J. C., Pollington A. D., On the range of fractional parts { ξ ( p / q ) n } , 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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.