The characterization of the minimal paths in the Christoffel tree according to a second-order balancedness

Andrea Frosini (1) , Lama Tarsissi (2)

1 DSMI - Department of Mathematics and Computer Science / Dipartimento di Scienze Matematiche e Informatiche "Roberto Magari"
2 LAMA - Laboratoire Analyse et de Mathématiques Appliquées

In 1842, Dirichlet observed that any real number α can be obtained as the limit of a sequence (pn qn) of irreducible rational numbers. Few years later, M.Stern (1858) and A.Brocot (1861) defined a tree-like arrangement of all the (irreducible) rational numbers whose infinite paths are the Dirichlet sequences of the real numbers and are characterized by their continued fraction representations. The Stern-Brocot tree is equivalent to the Christoffel tree obtained by ordering the Christoffel words according to their standard factorization. We first prove that the Fibonacci word's prefixes define minimal path in the Christoffel tree with respect to the second order balancedness parameter defined on Christoffel words. Switching back to the Stern-Brocot tree, we generalize this result and we obtain a characterization of all the minimal paths in terms of numerical suffixes of the continued fraction representation of the Christoffel words.

Domaines: Combinatoire [math.CO]; Théorie des nombres [math.NT]; Systèmes dynamiques [math.DS]

Andrea Frosini, Lama Tarsissi. The characterization of the minimal paths in the Christoffel tree according to a second-order balancedness. Developments in Language Theory-DLT2020, May 2020, Tampa, United States. ⟨hal-02495529⟩