Abstract β -expansions and ultimately periodic representations

Michel Rigo[1]; Wolfgang Steiner[2]

  • [1] Université de Liège, Institut de Mathématiques, Grande Traverse 12 (B 37), B-4000 Liège, Belgium.
  • [2] TU Wien, Institut für Diskrete Mathematik und Geometrie, Wiedner Hauptstrasse 8-10/104, A-1040 Wien, Austria Universität Wien, Institut für Mathematik, Strudlhofgasse 4, A-1090 Wien, Austria.

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 1, page 283-299
  • ISSN: 1246-7405

Abstract

top
For abstract numeration systems built on exponential regular languages (including those coming from substitutions), we show that the set of real numbers having an ultimately periodic representation is ℚ ( β ) if the dominating eigenvalue β > 1 of the automaton accepting the language is a Pisot number. Moreover, if β is neither a Pisot nor a Salem number, then there exist points in ℚ ( β ) which do not have any ultimately periodic representation.

How to cite

top

Rigo, Michel, and Steiner, Wolfgang. "Abstract $\beta $-expansions and ultimately periodic representations." Journal de Théorie des Nombres de Bordeaux 17.1 (2005): 283-299. <http://eudml.org/doc/249459>.

@article{Rigo2005,
abstract = {For abstract numeration systems built on exponential regular languages (including those coming from substitutions), we show that the set of real numbers having an ultimately periodic representation is $\mathbb\{Q\}(\beta )$ if the dominating eigenvalue $\beta &gt;1$ of the automaton accepting the language is a Pisot number. Moreover, if $\beta $ is neither a Pisot nor a Salem number, then there exist points in $\mathbb\{Q\}(\beta )$ which do not have any ultimately periodic representation.},
affiliation = {Université de Liège, Institut de Mathématiques, Grande Traverse 12 (B 37), B-4000 Liège, Belgium.; TU Wien, Institut für Diskrete Mathematik und Geometrie, Wiedner Hauptstrasse 8-10/104, A-1040 Wien, Austria Universität Wien, Institut für Mathematik, Strudlhofgasse 4, A-1090 Wien, Austria.},
author = {Rigo, Michel, Steiner, Wolfgang},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {-expansions; numeration systems; periodic expansions; Pisot numbers},
language = {eng},
number = {1},
pages = {283-299},
publisher = {Université Bordeaux 1},
title = {Abstract $\beta $-expansions and ultimately periodic representations},
url = {http://eudml.org/doc/249459},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Rigo, Michel
AU - Steiner, Wolfgang
TI - Abstract $\beta $-expansions and ultimately periodic representations
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 1
SP - 283
EP - 299
AB - For abstract numeration systems built on exponential regular languages (including those coming from substitutions), we show that the set of real numbers having an ultimately periodic representation is $\mathbb{Q}(\beta )$ if the dominating eigenvalue $\beta &gt;1$ of the automaton accepting the language is a Pisot number. Moreover, if $\beta $ is neither a Pisot nor a Salem number, then there exist points in $\mathbb{Q}(\beta )$ which do not have any ultimately periodic representation.
LA - eng
KW - -expansions; numeration systems; periodic expansions; Pisot numbers
UR - http://eudml.org/doc/249459
ER -

References

top
  1. A. Bertrand, Développements en base de Pisot et répartition modulo 1 . C. R. Acad. Sc. Paris 285 (1977), 419–421. Zbl0362.10040MR447134
  2. V. Bruyère, G. Hansel, Bertrand numeration systems and recognizability. Latin American Theoretical INformatics (Valparaíso, 1995). Theoret. Comput. Sci. 181 (1997), 17–43. Zbl0957.11015MR1463527
  3. J.-M. Dumont, A. Thomas, Systèmes de numération et fonctions fractales relatifs aux substitutions. J. Theoret. Comput. Sci. 65 (1989), 153–169. Zbl0679.10010MR1020484
  4. S. Eilenberg, Automata, languages, and machines. Vol. A, Pure and Applied Mathematics, Vol. 58, Academic Press , New York (1974). Zbl0317.94045MR530382
  5. C. Frougny, B. Solomyak, On representation of integers in linear numeration systems. In Ergodic theory of Z d actions (Warwick, 1993–1994), 345–368, London Math. Soc. Lecture Note Ser. 228, Cambridge Univ. Press, Cambridge (1996). Zbl0856.11007MR1411227
  6. C. Frougny, Numeration systems. In M. Lothaire, Algebraic combinatorics on words, Encyclopedia of Mathematics and its Applications 90. Cambridge University Press, Cambridge (2002). 
  7. P. B. A. Lecomte, M. Rigo, Numeration systems on a regular language. Theory Comput. Syst. 34 (2001), 27–44. Zbl0969.68095MR1799066
  8. P. Lecomte, M. Rigo, On the representation of real numbers using regular languages. Theory Comput. Syst. 35 (2002), 13–38. Zbl0993.68050MR1879170
  9. P. Lecomte, M. Rigo, Real numbers having ultimately periodic representations in abstract numeration systems. Inform. and Comput. 192 (2004), 57–83. Zbl1055.11005MR2063624
  10. W. Parry, On the β -expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401–416. Zbl0099.28103
  11. A. Rényi, Representation for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493. Zbl0079.08901MR97374
  12. K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc. 12 (1980), 269–278. Zbl0494.10040MR576976

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.