The (u,v)-Calkin–Wilf forest

Han, Sandie; Masuda, Ariane M.; Singh, Satyanand; Thiel, Johann

doi:10.1142/s1793042116500809

Cited by 7 publications

(14 citation statements)

References 17 publications

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“…z+2 is the root of the PLFT CWtree containing 7z+8 4z+5 . The continued fractions of a positive rational number and its children in the Calkin-Wilf tree are closely related [8,14]. A similar result holds for PLFT CW-trees.…”

Section: Example 1 Consider the Plft 7z+8mentioning

confidence: 58%

Orphans in Forests of Linear Fractional Transformations

Han

Masuda

Singh

et al. 2016

Electron. J. Combin.

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The set of positive linear fractional transformations (PLFTs) is partitioned into an infinite forest of PLFT Calkin-Wilf-trees. The roots of these trees are called orphans. In this paper, we provide a combinatorial formula for the number of orphan PLFTs with fixed determinant D. Then we provide a way of determining the orphan of a PLFT Calkin-Wilf-tree for a given PLFT. In addition, we show that every positive complex number is the descendant of a complex (u, v)orphan. Date

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Section: Example 1 Consider the Plft 7z+8mentioning

confidence: 58%

Orphans in Forests of Linear Fractional Transformations

Han

Masuda

Singh

et al. 2016

Electron. J. Combin.

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“…Taking the limit as n → ∞ of both sides of (5) shows that, to complete the proof, it is enough to prove that lim n→∞ 1 2 n n k=1 y∈T (n−k)…”

Section: Resultsmentioning

confidence: 99%

“…More generally, the trees T (u,v) (z) form a partition of Q + as z runs over [1/u, v] ∩ Q; see [8]. The Calkin-Wilf tree has many other interesting properties [3,5,6,8,9], one of which is the fact that the mean value of vertices of depth n converges to 3/2 as n → ∞ [1,10]. Our main result generalizes this property for all (u, v)-Calkin-Wilf trees.…”

Section: Introductionmentioning

confidence: 99%

Mean Row Values in (u, v)-Calkin–Wilf Trees

Han¹,

Masuda²,

Singh³

et al. 2019

Combinatorial and Additive Number Theory III

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We fix integers u, v ≥ 1, and consider an infinite binary tree T (u,v) (z) with a root node whose value is a positive rational number z. For every vertex a/b, we label the left child as a/(ua + b) and right child as (a + vb)/b. The resulting tree is known as the (u, v)-Calkin-Wilf tree. As z runs over [1/u, v] ∩ Q, the vertex sets of T (u,v) (z) form a partition of Q + . When u = v = 1, the mean row value converges to 3/2 as the row depth increases. Our goal is to extend this result for any u, v ≥ 1. We show that, when z ∈ [1/u, v] ∩ Q, the mean row value in T (u,v) (z) converges to a value close to v + log 2/u uniformly on z.

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“…Our main tool in the next section will be the following lemma, which is a straightforward generalization of [4,Lemma 5] and is similar to [ The method of proof for Proposition 3 is nearly identical to that of Proposition 1 using Lemma 5 part (a).…”

Section: Letmentioning

confidence: 99%

Subgroups of 𝑆𝐿₂(ℤ) characterized by certain continued fraction representations

Han¹,

Masuda²,

Singh³

et al. 2020

Proc. Amer. Math. Soc.

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For positive integers u and v, let Lu = 1 0 u 1 and Rv = 1 v 0 1 . Let Su,v be the monoid generated by Lu and Rv, and Gu,v be the group generated by Lu and Rv. In this paper we expand on a characterization of matrices M = a b c d in S k,k and G k,k when k ≥ 2 given by Esbelin and Gutan to Su,v when u, v ≥ 2 and Gu,v when u, v ≥ 3. We give a simple algorithmic way of determining if M is in Gu,v using a recursive function and the short continued fraction representation of b/d.

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The (u,v)-Calkin–Wilf forest

Cited by 7 publications

References 17 publications

Orphans in Forests of Linear Fractional Transformations

Orphans in Forests of Linear Fractional Transformations

Mean Row Values in (u, v)-Calkin–Wilf Trees

Subgroups of 𝑆𝐿₂(ℤ) characterized by certain continued fraction representations

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