Orphans in Forests of Linear Fractional Transformations

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

doi:10.37236/5684

Cited by 3 publications

(3 citation statements)

References 15 publications

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“…[1,3,4,7,9,14,15,16,19]). For work related to this paper, see [11,12]. The first four rows the Calkin-Wilf tree are as follows:…”

Section: The Calkin-wilf Tree Of Rational Numbersmentioning

confidence: 99%

Free monoids and forests of rational numbers

Nathanson

2017

Discrete Applied Mathematics

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Abstract. The Calkin-Wilf tree is an infinite binary tree whose vertices are the positive rational numbers. Each such number occurs in the tree exactly once and in the form a/b, where are a and b are relatively prime positive integers. This tree is associated with the matrices L 1 = 1 0 1 1 and, which freely generate the monoid SL 2 (N 0 ) of 2 × 2 matrices with determinant 1 and nonnegative integral coordinates. For other pairs of matrices Lu and Rv that freely generate submonoids of GL 2 (N 0 ), there are forests of infinitely many rooted infinite binary trees that partition the set of positive rational numbers, and possess a remarkable symmetry property.

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“…[1,3,4,7,9,14,15,16,19]). For work related to this paper, see [11,12]. The first four rows the Calkin-Wilf tree are as follows:…”

Section: The Calkin-wilf Tree Of Rational Numbersmentioning

confidence: 99%

Free monoids and forests of rational numbers

Nathanson

2017

Discrete Applied Mathematics

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“…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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“…To reduce some of our calculations and to better organize and present our work, we use a generalization of the Calkin-Wilf tree [4] for positive linear fractional transformations (PLFTs) due to Nathanson [10]. In particular, we will use the matrix version 2 of this tree (see [4,5,6,10] for a more thorough history of this material).…”

Section: Introductionmentioning

confidence: 99%

Maximal entries of elements in certain matrix monoids

Han¹,

Masuda²,

Singh³

et al. 2017

Preprint

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Let Lu = 1 0 u 1 and Rv = 1 v 0 1 be matrices in SL2(Z) with u, v ≥ 1. Since the monoid generated by Lu and Rv is free, we can associate a depth to each element based on its product representation. In the cases where u = v = 2 and u = v = 3, Bromberg, Shpilrain, and Vdovina determined the depth n matrices containing the maximal entry for each n ≥ 1. By using ideas from our previous work on (u, v)-Calkin-Wilf trees, we extend their results for any u, v ≥ 1 and in the process we recover the Fibonacci and some Lucas sequences. As a consequence we obtain bounds which guarantee collision resistance on a family of hashing functions based on Lu and Rv.

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Orphans in Forests of Linear Fractional Transformations

Cited by 3 publications

References 15 publications

Free monoids and forests of rational numbers

Free monoids and forests of rational numbers

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

Maximal entries of elements in certain matrix monoids

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