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 treelike 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 remark that the Fibonacci word’s prefixes belong to a minimal path in the Christoffel tree with respect to the second order balancedness parameter defined on Christoffel words. This alows us to switch back to the Stern-Brocot tree, in order to give a characterization of the continued fraction representation for all the rational numbers belonging to minimal paths with respect to the growth of the second order balancedness.
The Characterization of Rational Numbers Belonging to a Minimal Path in the Stern-Brocot Tree According to a Second Order Balancedness / Andrea Frosini, Lama Tarsissi. - ELETTRONICO. - 12086:(2020), pp. 319-331. (Intervento presentato al convegno 24th International Conference on Developments in Language Theory, DLT 2020 tenutosi a usa nel 2020) [10.1007/978-3-030-48516-0_24].
The Characterization of Rational Numbers Belonging to a Minimal Path in the Stern-Brocot Tree According to a Second Order Balancedness
Andrea Frosini;Lama Tarsissi
2020
Abstract
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 treelike 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 remark that the Fibonacci word’s prefixes belong to a minimal path in the Christoffel tree with respect to the second order balancedness parameter defined on Christoffel words. This alows us to switch back to the Stern-Brocot tree, in order to give a characterization of the continued fraction representation for all the rational numbers belonging to minimal paths with respect to the growth of the second order balancedness.File | Dimensione | Formato | |
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