Linear fractional transformations of continued fractions with bounded partial quotients

J. C. Lagarias; J. O. Shallit

Displaying similar documents to “Linear fractional transformations of continued fractions with bounded partial quotients”

Leaping convergents of Tasoev continued fractions

Takao Komatsu (2011)

Discussiones Mathematicae - General Algebra and Applications

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Denote the n-th convergent of the continued fraction by pₙ/qₙ = [a₀;a₁,...,aₙ]. We give some explicit forms of leaping convergents of Tasoev continued fractions. For instance, [0;ua,ua²,ua³,...] is one of the typical types of Tasoev continued fractions. Leaping convergents are of the form $p_{r n + i} / q_{r n + i}$ (n=0,1,2,...) for fixed integers r ≥ 2 and 0 ≤ i ≤ r-1.

Leaping convergents of Hurwitz continued fractions

Takao Komatsu (2011)

Discussiones Mathematicae - General Algebra and Applications

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Let pₙ/qₙ = [a₀;a₁,...,aₙ] be the n-th convergent of the continued fraction expansion of [a₀;a₁,a₂,...]. Leaping convergents are those of every r-th convergent $p_{r n + i} / q_{r n + i}$ (n = 0,1,2,...) for fixed integers r and i with r ≥ 2 and i = 0,1,...,r-1. The leaping convergents for the e-type Hurwitz continued fractions have been studied. In special, recurrence relations and explicit forms of such leaping convergents have been treated. In this paper, we consider recurrence relations and explicit forms...

Symmetry and folding of continued fractions

Alfred J. Van der Poorten (2002)

Journal de théorie des nombres de Bordeaux

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Michel Mendès France's “Folding Lemma” for continued fraction expansions provides an unusual explanation for the well known symmetry in the expansion of a quadratic irrational integer.

Automatic continued fractions are transcendental or quadratic

Yann Bugeaud (2013)

Annales scientifiques de l'École Normale Supérieure

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We establish new combinatorial transcendence criteria for continued fraction expansions. Let $α = [0; a_{1}, a_{2}, ...]$ be an algebraic number of degree at least three. One of our criteria implies that the sequence of partial quotients ${(a_{ℓ})}_{ℓ \geq 1}$ of $α$ is not ‘too simple’ (in a suitable sense) and cannot be generated by a finite automaton.

The best Diophantine approximation functions by continued fractions

Jing Cheng Tong (1996)

Mathematica Bohemica

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Let $ξ = [a_{0}; a_{1}, a_{2}, \dots, a_{i}, \dots]$ be an irrational number in simple continued fraction expansion, $p_{i} / q_{i} = [a_{0}; a_{1}, a_{2}, \dots, a_{i}]$ , $M_{i} = q_{i}^{2} | ξ - p_{i} / q_{i} |$ . In this note we find a function $G (R, r)$ such that $\begin{matrix} M_{n + 1} < R and M_{n - 1} < r imply M_{n} > G (R, r), & M_{n + 1} > R and M_{n - 1} > r imply M_{n} < G (R, r) . \end{matrix}$ Together with a result the author obtained, this shows that to find two best approximation functions $\tilde{H} (R, r)$ and $\tilde{L} (R, r)$ is a well-posed problem. This problem has not been solved yet.

Algebraic continued fractions in $_{q} ((T^{- 1}))$ and recurrent sequences in $_{q}$

Alain Lasjaunias (2008)

Acta Arithmetica

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Legendre polynomials and supercongruences

Zhi-Hong Sun (2013)

Acta Arithmetica

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Let p > 3 be a prime, and let Rₚ be the set of rational numbers whose denominator is not divisible by p. Let Pₙ(x) be the Legendre polynomials. In this paper we mainly show that for m,n,t ∈ Rₚ with m≢ 0 (mod p), $P_{[p / 6]} (t) \equiv - (3 / p) \sum_{x = 0}^{p - 1} ((x ³ - 3 x + 2 t) / p) (m o d p)$ and $(\sum_{x = 0}^{p - 1} ((x ³ + m x + n) / p)) ² \equiv ((- 3 m) / p) \sum_{k = 0}^{[p / 6]} (\binom{2 k}{k}) (\binom{3 k}{k}) (\binom{6 k}{3 k}) {((4 m ³ + 27 n ²) / (12 ³ \cdot 4 m ³))}^{k} (m o d p)$ , where (a/p) is the Legendre symbol and [x] is the greatest integer function. As an application we solve some conjectures of Z. W. Sun and the author concerning $\sum_{k = 0}^{p - 1} (\binom{2 k}{k}) (\binom{3 k}{k}) (\binom{6 k}{3 k}) / m^{k} (m o d p ²)$ , where m is an integer not divisible by p.

On the length of rational continued fractions over $_{q} (X)$

S. Driss (2015)

Discussiones Mathematicae - General Algebra and Applications

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Let $_{q}$ be a finite field and $A (Y) \in_{q} (X, Y)$ . The aim of this paper is to prove that the length of the continued fraction expansion of $A (P); P \in_{q} [X]$ , is bounded.

On the length of the continued fraction for values of quotients of power sums

Pietro Corvaja, Umberto Zannier (2005)

Journal de Théorie des Nombres de Bordeaux

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Generalizing a result of Pourchet, we show that, if $α, β$ are power sums over $ℚ$ satisfying suitable necessary assumptions, the length of the continued fraction for $α (n) / β (n)$ tends to infinity as $n \to \infty$ . This will be derived from a uniform Thue-type inequality for the rational approximations to the rational numbers $α (n) / β (n)$ , $n \in ℕ$ .

A new class of continued fraction expansions

Cor Kraaikamp (1991)

Acta Arithmetica

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On approximation by Lüroth series

Karma Dajani, Cor Kraaikamp (1996)

Journal de théorie des nombres de Bordeaux

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Let $x \in] 0, 1]$ and $p_{n} / q_{n}, n \geq 1$ be its sequence of Lüroth Series convergents. Define the approximation coefficients $θ_{n} = θ_{n} (x)$ by $q_{n} x - p_{n}, n \geq 1$ . In [BBDK] the limiting distribution of the sequence ${(θ_{n})}_{n \geq 1}$ was obtained for a.e. $x$ using the natural extension of the ergodic system underlying the Lüroth Series expansion. Here we show that this can be done without the natural extension. In fact we will prove that for each $n, θ_{n}$ is already distributed according to the limiting distribution. Using the natural extension we will study the distribution...