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Shin-ichi Yasutomi

Publications and source records attributed to Shin-ichi Yasutomi.

11 recordsLinked to original sources
arXiv2026

On the Ordering and Injectivity of Lagrange Constants Associated with Certain Substitutions

Motivated by the modern characterization of the Markoff spectrum via mechanical words and the Markoff Uniqueness Conjecture, we study the Lagrange constants of continued fractions generated by a parameterized family of substitutions $ϕ_{a,b}$ $ϕ_{a,b}$ defined by $0 \mapsto aa$ and $1 \mapsto bb$ with $b = a + 1$. Based on a three-fold arithmetic classification of pairs of rational numbers $0 \le x < y \le 1$, we investigate the order relations between the corresponding Lagrange constants $\mathcal{L}([ϕ_{a,a+1}(G(x))])$ associated with the mechanical words $G(x)$. While an inequality is established for pairs of the second type (type (2)) whenever $a \ge 1$, the comparisons for type (1) and type (3) pairs hold for $a \ge 2$. Consequently, for any integer $a \ge 2$, where the order relations across all three types are completely determined, we establish an analogue of the Markoff Uniqueness Conjecture: the map $x \mapsto \mathcal{L}([ϕ_{a,a+1}(G(x))])$ is injective on $\mathbb{Q} \cap [0, 1]$. Furthermore, supported by extensive numerical computations, we propose a general conjecture on the order relations for arbitrary $b > a$, highlighting a sharp phase transition in ordering behavior at the boundary $b = a^2 + 2$.

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arXiv2024

Periodic points in complex continued fractions by J. Hurwitz

J. Hurwitz introduced an algorithm that generates a continued fraction expansion for complex numbers $α\in \mathbb{C}$, where the partial quotients belong to $(1+i)\mathbb{Z}[i]$. J. Hurwitz's work also provides a result analogous to Lagrange's theorem on periodic continued fractions, describing purely periodic points using the dual continued fraction expansion. S. Tanaka \cite{ST} examined the identical algorithm and constructed the natural extension of the transformation generating the continued fraction expansion and established its ergodic properties. In this paper, we aim to describe J. Hurwitz's insights into purely periodic points explicitly through the natural extension.

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arXiv2024

Star-shaped trajectories of certain billiards around a triangle

We explore the triangle outer billiards map in points at infinity in the hyperbolic plane, focusing on the rotation number. Building on Dogru and Tabachnikov's work, which established the conditions for triangles where the rotation number of the billiard map is $1/3$, we examine cases where the rotation number is $2/5$. We provide a sufficient condition for this rotation number and show its necessity for large isosceles triangles. The results are framed within the context of the Beltrami-Klein model. We concludes with a conjecture based on the findings.

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arXiv2023

Billiards in a circle with trajectories circumscribing a triangle

Dogru and Tabachnikov in 2003 explored the polygonal outer billiard map in the hyperbolic plane and introduced a class of convex polygons called 'large'. They particularly sought conditions for a triangle to be classified as large. For a large triangle, there exist two triangles that are circumscribed around it and inscribed within the unit circle. In the Klein-Beltrami model of hyperbolic geometry, we reformulate the conditions for a triangle to be classified as 'large' in a more Euclidean geometric manner. A proposed measure of its Euclidean geometric size when the triangle is considered 'large' is introduced, and an evaluation of this measure is conducted. We also provide an explicit formula for an isosceles triangle.

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arXiv2023

Simultaneous Convergent Continued Fraction Algorithm for Real and $p$-adic Fields with Applications to Quadratic Fields

Let $p$ be a prime number and $K$ be a field with embeddings into $\mathbb{R}$ and $\mathbb{Q}_p$. We propose an algorithm that generates continued fraction expansions converging in $\mathbb{Q}_p$ and is expected to simultaneously converge in both $\mathbb{R}$ and $\mathbb{Q}_p$. This algorithm produces finite continued fraction expansions for rational numbers. In the case of $p=2$ and if $K$ is a quadratic field, the continued fraction expansions generated by this algorithm converge in $\mathbb{R}$, and they are eventually periodic or finite. For an element $α$ in $K$, let $p_n/q_n$ denote the $n$-th convergent. There exist constants $u_1$ and $u_2$ in ${\mathbb R}_{>0}$ with $u_1 + u_2 = 2$, and constants $C_1$ and $C_2$ in ${\mathbb R}_{>0}$ such that $|α- p_n/q_n| < C_1/|q_n|^{u_1}$ and $|α- p_n/q_n|_2 < C_2/|q_n|^{u_2}$. Here, $|\cdot|_2$ represents the $2$-adic distance. For prime numbers $p > 2$, we present numerical experiences.

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arXiv2023

Arithmetical independence of certain uniform sets of algebraic integers

We study four (families of) sets of algebraic integers of degree less than or equal to three. Apart from being simply defined, we show that they share two distinctive characteristics: almost uniformity and arithmetical independence. Here, ``almost uniformity'' means that the elements of a finite set are distributed almost equidistantly in the unit interval, while ``arithmetical independence'' means that the number fields generated by the elements of a set do not have a mutual inclusion relation each other. Furthermore, we reveal to what extent the algebraic number fields generated by the elements of the four sets can cover quadratic or cubic fields.

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arXiv2021

Ergodicity for $p$-adic continued fraction algorithms

Following Schweiger's generalization of multidimensional continued fraction algorithms, we consider a very large family of $p$-adic multidimensional continued fraction algorithms, which include Schneider's algorithm, Ruban's algorithms, and the $p$-adic Jacobi-Perron algorithm as special cases. The main result is to show that all the transformations in the family are ergodic with respect to the Haar measure.

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arXiv2017

Continued fraction algorithms and Lagrange's theorem in ${\mathbb Q}_p$

We present several continued fraction algorithms, each of which gives an eventually periodic expansion for every quadratic element of ${\mathbb Q}_p$ over ${\mathbb Q}$ and gives a finite expansion for every rational number. We also give, for each of our algorithms, the complete characterization of elements having purely periodic expansions.

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