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

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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

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

First available in Project Euclid: 20 December 2012

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Zentralblatt MATH identifier

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]

Fibonacci numeration system words Fibonacci substitution adic representation


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.

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