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 (1) , Laurent Vivier (2)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 LDAR (EA_4434) - Laboratoire de Didactique André Revuz

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 give a new result on factors of the Fibonacci word $abaababaabaab\ldots$. Second, we present an expression of the sequence A004146 of \cite{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 particuler that it contains exactly $q$ ${\mathcal F}$-adic elements.

Mots clés: Fibonacci numeration system; Words; Fibonacci substitution; Adic representation

Domaines: Théorie des nombres [math.NT]

Benoît Rittaud, Laurent Vivier. Circular words and three applications: factors of the Fibonacci word, ${\mathcal F}$-adic numbers, and the sequence $1$, $5$, $16$, $45$, $121$, $320$,\ldots. 2011. ⟨hal-00566314⟩