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

In 1842, Dirichlet observed that any real number [Formula: see text] can be obtained as the limit of a sequence [Formula: see text] 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 i...

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Autores principales: Frosini, Andrea, Tarsissi, Lama
Formato: Online Artículo Texto
Lenguaje:English
Publicado: 2020
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Article
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7247880/
http://dx.doi.org/10.1007/978-3-030-48516-0_24
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author Frosini, Andrea
Tarsissi, Lama
author_facet Frosini, Andrea
Tarsissi, Lama
author_sort Frosini, Andrea
collection PubMed
description In 1842, Dirichlet observed that any real number [Formula: see text] can be obtained as the limit of a sequence [Formula: see text] 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.
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spelling pubmed-72478802020-05-26 The Characterization of Rational Numbers Belonging to a Minimal Path in the Stern-Brocot Tree According to a Second Order Balancedness Frosini, Andrea Tarsissi, Lama Developments in Language Theory Article In 1842, Dirichlet observed that any real number [Formula: see text] can be obtained as the limit of a sequence [Formula: see text] 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. 2020-05-26 /pmc/articles/PMC7247880/ http://dx.doi.org/10.1007/978-3-030-48516-0_24 Text en © Springer Nature Switzerland AG 2020 This article is made available via the PMC Open Access Subset for unrestricted research re-use and secondary analysis in any form or by any means with acknowledgement of the original source. These permissions are granted for the duration of the World Health Organization (WHO) declaration of COVID-19 as a global pandemic.
spellingShingle Article
Frosini, Andrea
Tarsissi, Lama
The Characterization of Rational Numbers Belonging to a Minimal Path in the Stern-Brocot Tree According to a Second Order Balancedness
title The Characterization of Rational Numbers Belonging to a Minimal Path in the Stern-Brocot Tree According to a Second Order Balancedness
title_full The Characterization of Rational Numbers Belonging to a Minimal Path in the Stern-Brocot Tree According to a Second Order Balancedness
title_fullStr The Characterization of Rational Numbers Belonging to a Minimal Path in the Stern-Brocot Tree According to a Second Order Balancedness
title_full_unstemmed The Characterization of Rational Numbers Belonging to a Minimal Path in the Stern-Brocot Tree According to a Second Order Balancedness
title_short The Characterization of Rational Numbers Belonging to a Minimal Path in the Stern-Brocot Tree According to a Second Order Balancedness
title_sort characterization of rational numbers belonging to a minimal path in the stern-brocot tree according to a second order balancedness
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7247880/
http://dx.doi.org/10.1007/978-3-030-48516-0_24
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