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The n -dimensional Stern-Brocot tree
Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
2019 (English)In: International Journal of Number Theory, ISSN 1793-0421, Vol. 15, no 6, p. 1219-1236Article in journal (Refereed) Published
Abstract [en]

This paper generalizes the Stern-Brocot tree to a tree that consists of all sequences of n coprime positive integers. As for n = 2, each sequence P is the sum of a specific set of other coprime sequences, its Stern-Brocot set B(P), where |B(P)| is the degree of P. With an orthonormal base as the root, the tree defines a fast iterative structure on the set of distinct directions in ℤ+n and a multiresolution partition of S+n-1. Basic proofs rely on a matrix representation of each coprime sequence, where the Stern-Brocot set forms the matrix columns. This induces a finitely generated submonoid SB(n, ℕ) of SL(n, ℕ), and a unimodular multidimensional continued fraction algorithm, also generalizing n = 2. It turns out that the n-dimensional subtree starting with a sequence P is isomorphic to the entire |B(P)|-dimensional tree. This allows basic combinatorial properties to be established. It turns out that also in this multidimensional version, Fibonacci-type sequences have maximal sequence sum in each generation. © 2019 World Scientific Publishing Company.

Place, publisher, year, edition, pages
World Scientific Publishing Co. Pte Ltd , 2019. Vol. 15, no 6, p. 1219-1236
Keywords [en]
coprimality, matrix representation, multidimensional continued fraction, S L (n, ℕ), Stern-Brocot tree
National Category
Geometry Mathematical Analysis
Identifiers
URN: urn:nbn:se:bth-17772DOI: 10.1142/S1793042119500672ISI: 000476732000007Scopus ID: 2-s2.0-85062921934OAI: oai:DiVA.org:bth-17772DiVA, id: diva2:1302689
Available from: 2019-04-05 Created: 2019-04-05 Last updated: 2019-09-10Bibliographically approved

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Lennerstad, Håkan

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