Exponents of Diophantine Approximation and Sturmian Continued Fractions

Yann Bugeaud; Michel Laurent

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Continued fractions, multidimensional diophantine approximations and applications

Nikolai G. Moshchevitin (1999)

Journal de théorie des nombres de Bordeaux

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This paper is a brief review of some general Diophantine results, best approximations and their applications to the theory of uniform distribution.

A function from Diophantine approximations.

Shiu, P. (1999)

Publications de l'Institut Mathématique. Nouvelle Série

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Best simultaneous diophantine approximations of some cubic algebraic numbers

Nicolas Chevallier (2002)

Journal de théorie des nombres de Bordeaux

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Let $α$ be a real algebraic number of degree $3$ over $ℚ$ whose conjugates are not real. There exists an unit $ζ$ of the ring of integer of $K = ℚ (α)$ for which it is possible to describe the set of all best approximation vectors of $θ = (ζ, ζ^{2})$ .’

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.

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]. ...

A note on linear recurrent Mahler numbers.

Barat, Guy, Frougny, Christiane, Pethő, Attila (2005)

Integers

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Square roots with many good approximants.

Dujella, Andrej, Petričević, Vinko (2005)

Integers

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Diophantine equations with linear recurrences An overview of some recent progress

Umberto Zannier (2005)

Journal de Théorie des Nombres de Bordeaux

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We shall discuss some known problems concerning the arithmetic of linear recurrent sequences. After recalling briefly some longstanding questions and solutions concerning zeros, we shall focus on recent progress on the so-called “quotient problem” (resp. " $d$ -th root problem"), which in short asks whether the integrality of the values of the quotient (resp. $d$ -th root) of two (resp. one) linear recurrences implies that this quotient (resp. $d$ -th root) is itself a recurrence. We shall also...