The Characterization of Rational Numbers Belonging to a Minimal Path in the Stern-Brocot Tree According to a Second Order Balancedness

Autor:	Andrea Frosini, Lama Tarsissi
Rok vydání:	2020
Předmět:	Combinatorics Christoffel symbols Tree (descriptive set theory) Rational number Stern–Brocot tree Limit of a sequence Order (ring theory) Fibonacci word Computer Science::Formal Languages and Automata Theory Mathematics Real number
Zdroj:	Developments in Language Theory ISBN: 9783030485153 DLT
DOI:	10.1007/978-3-030-48516-0_24
Popis:	In 1842, Dirichlet observed that any real number \(\alpha \) can be obtained as the limit of a sequence \((\frac{p_n}{q_n})\) 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 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.
Databáze:	OpenAIRE
Externí odkaz:	https://explore.openaire.eu/search/publication?articleId=doi_________::400d5028e62e263f8e5beb14819d6dc9 https://doi.org/10.1007/978-3-030-48516-0_24 Zobrazit plný text záznamu