Displaying similar documents to “The best Diophantine approximation functions by continued fractions”

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.

Diophantine approximation on Veech surfaces

Pascal Hubert, Thomas A. Schmidt (2012)

Bulletin de la Société Mathématique de France

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We show that Y. Cheung’s general Z -continued fractions can be adapted to give approximation by saddle connection vectors for any compact translation surface. That is, we show the finiteness of his Minkowski constant for any compact translation surface. Furthermore, we show that for a Veech surface in standard form, each component of any saddle connection vector dominates its conjugates in an appropriate sense. The saddle connection continued fractions then allow one to recognize certain...

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...

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 → ∞ . This will be derived from a uniform Thue-type inequality for the rational approximations to the rational numbers α ( n ) / β ( n ) , n ∈ ℕ .

Arithmetic diophantine approximation for continued fractions-like maps on the interval

Avraham Bourla (2014)

Acta Arithmetica

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We establish arithmetical properties and provide essential bounds for bi-sequences of approximation coefficients associated with the natural extension of maps, leading to continued fraction-like expansions. These maps are realized as the fractional part of Möbius transformations which carry the end points of the unit interval to zero and infinity, extending the classical regular and backwards continued fraction expansions.

Tong’s spectrum for Rosen continued fractions

Cornelis Kraaikamp, Thomas A. Schmidt, Ionica Smeets (2007)

Journal de Théorie des Nombres de Bordeaux

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In the 1990s, J.C. Tong gave a sharp upper bound on the minimum of k consecutive approximation constants for the nearest integer continued fractions. We generalize this to the case of approximation by Rosen continued fraction expansions. The Rosen fractions are an infinite set of continued fraction algorithms, each giving expansions of real numbers in terms of certain algebraic integers. For each, we give a best possible upper bound for the minimum in appropriate consecutive blocks of...

Introduction to Diophantine Approximation

Yasushige Watase (2015)

Formalized Mathematics

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In this article we formalize some results of Diophantine approximation, i.e. the approximation of an irrational number by rationals. A typical example is finding an integer solution (x, y) of the inequality |xθ − y| ≤ 1/x, where 0 is a real number. First, we formalize some lemmas about continued fractions. Then we prove that the inequality has infinitely many solutions by continued fractions. Finally, we formalize Dirichlet’s proof (1842) of existence of the solution [12], [1]. ...

Linear fractional transformations of continued fractions with bounded partial quotients

J. C. Lagarias, J. O. Shallit (1997)

Journal de théorie des nombres de Bordeaux

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Let θ be a real number with continued fraction expansion θ = a 0 , a 1 , a 2 , ⋯ , and let M = a b c d be a matrix with integer entries and nonzero determinant. If θ has bounded partial quotients, then a θ + b c θ + d = a 0 * , a 1 * , a 2 * , ⋯ also has bounded partial quotients. More precisely, if a j ≤ K for all sufficiently large j , then a j * ≤ | det ( M ) | ( K + 2 ) for all sufficiently large j . We also give a weaker bound valid for all a j * with j ≥ 1 . The proofs use the homogeneous Diophantine approximation constant L ∞ θ = lim sup q → ∞ q q θ - 1 . We show that ...