Functiones et Approximatio Commentarii Mathematici

Circular words and three applications: factors of the Fibonacci word, $\mathcal F$-adic numbers, and the sequence 1, 5, 16, 45, 121, 320,\ldots

Benoît Rittaud and Laurent Vivier

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We introduce the notion of {\em circular words} with a combinatorial constraint derived from the Zeckendorf (Fibonacci) numeration system, and get explicit group structures for these words. As a first application, we establish a new result on factors of the Fibonacci word $abaababaabaab\ldots$. Second, we present an expression of the sequence A004146 of [Sloane] in terms of a product of expressions involving roots of unity. Third, we consider the equivalent of $p$-adic numbers that arise by the use of the numeration system defined by the Fibonacci sequence instead of the usual numeration system in base $p$. Among such {\em ${\mathcal F}$-adic numbers}, we give a~characterization of the subset of those which are {\em rational} (that is: a root of an equation of the form $qX=p$, for integral values of $p$ and $q$) by a periodicity property. Eventually, with the help of circular words, we give a complete description of the set of roots of $qX=p$, showing in particular that it contains exactly $q$ ${\mathcal F}$-adic elements.

Article information

Source
Funct. Approx. Comment. Math. Volume 47, Number 2 (2012), 207-231.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
http://projecteuclid.org/euclid.facm/1356012916

Digital Object Identifier
doi:10.7169/facm/2012.47.2.6

Mathematical Reviews number (MathSciNet)
MR3051449

Zentralblatt MATH identifier
1260.68313

Subjects
Primary: 68R15: Combinatorics on words
Secondary: 11A63: Radix representation; digital problems {For metric results, see 11K16} 11E95: $p$-adic theory 37B10: Symbolic dynamics [See also 37Cxx, 37Dxx]

Keywords
Fibonacci numeration system words Fibonacci substitution adic representation

Citation

Rittaud, Benoît; Vivier, Laurent. Circular words and three applications: factors of the Fibonacci word, $\mathcal F$-adic numbers, and the sequence 1, 5, 16, 45, 121, 320,\ldots. Funct. Approx. Comment. Math. 47 (2012), no. 2, 207--231. doi:10.7169/facm/2012.47.2.6. http://projecteuclid.org/euclid.facm/1356012916.


Export citation

References

  • P. Ambro\uz and Ch. Frougny, On alpha-adic expansions in Pisot bases, Theoret. Comput. Sci. 380 (2007), 238–250.
  • G. Barat and P. Liardet, Dynamical systems originated in the Ostrowski alpha-expansion, Ann. Univ. Sci. Budapest Sect. Comput. 24 (2004), 133–184.
  • Ch. Frougny and J. Sakarovitch, Two groups associated with quadratic Pisot units, Internat. J. Algebra Comput 12 (2002), 825–847.
  • P. Grabner, P. Liardet and R. Tichy, Odometers and systems of numeration, Acta Arith. LXX (1995), 103–123.
  • M. Lothaire, Combinatorics on words, Cambridge University Press, Cambridge, 1997.
  • K. Rebman, The sequence: $1$ $5$ $16$ $45$ $121$ $320\ldots$ in combinatorics, Fibonacci Quart. 13 (1975), 51–55.
  • N. Sloane, http://oeis.org/A004146
  • K. Schmidt, Algebraic codings of expansive group automorphisms and two-sided beta-shifts, Monatsh. Math 129 (2000), 37-61.
  • N. Sidorov, An arithmetic group associated with a Pisot unit, and its symbolic-dynamical representation, Acta Arith. 101 (2002), 199–213.
  • N. Sidorov and A. Vershik, Ergodic properties of the Erdős measure, the entropy of the golden shift, and related problems, Monatsh. Math. 126 (1998), 215-261.
  • E. Zeckendorf, Représentation des nombres naturels par une somme de nombres de Fibonacci ou de nombres de Lucas, Bull. Soc. Roy. Sci. Liège 41 (1972), 179–182.