Diophantine Approximation and Continued Fraction Expansions of Algebraic Power Series in Positive Characteristic

Lasjaunias, Alain

doi:10.1006/jnth.1997.2160

Cited by 19 publications

(17 citation statements)

References 8 publications

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“…The continued fraction expansion for this root α in F(3), calculated by computer, was conjectured in [9]. Some ten years later this conjecture was proved in [2] and also, shortly afterwards, with a different method in [6]. Here we have α = [0, a 1 , a 2 , .…”

Section: Introductionmentioning

confidence: 91%

On a two-valued sequence and related continued fractions in power series fields

2017

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We explicitly describe a noteworthy transcendental continued fraction in the field of power series over Q, having irrationality measure equal to 3. This continued fraction is a generating function of a particular sequence in the set {1, 2}. The origin of this sequence, whose study was initiated in a recent paper, is to be found in another continued fraction, in the field of power series over F 3 , which satisfies a simple algebraic equation of degree 4, introduced thirty years ago by D. Robbins.

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Section: Introductionmentioning

confidence: 91%

On a two-valued sequence and related continued fractions in power series fields

2017

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“…Buck and Robbins established this conjecture in [6]. Shortly after another proof of this conjecture was given in [15]. We have α = [0, a 1 , a 2 , .…”

Section: A Substitutive But Not Automatic Sequencementioning

confidence: 97%

“…The fact that α is not hyperquadratic was proved in [15] (see the remark after Theorem A, p.209). Indeed the knowledge of the continued fraction allows to show that the irrationality measure is equal to 2.…”

Section: A Substitutive But Not Automatic Sequencementioning

confidence: 99%

“…These hyperquadratic continued fractions were studied independently by Schmidt [21] and Thakur [22] (particularly for ε = 1). Let us recall that the elements in H(q), called here hyperquadratic, were first named in [15] as algebraic of class I, and then the elements studied by Schmidt and Thakur were called algebraic of class IA. The possibility of choosing arbitrarily the vector A ℓ has an important consequence.…”

Section: Three Families Of Hyperquadratic Continued Fractionsmentioning

confidence: 99%

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Hyperquadratic continued fractions and automatic sequences

Lasjaunias

Yao

2016

Finite Fields and Their Applications

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“…Introduction. Continued fractions [8,14] are a useful tool in many number theoretical problems and in numerical computing. It is well known that the simple continued fraction expansion of a single real number gives the best solution to its rational approximation problem.…”

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confidence: 99%

Multi-continued fraction algorithm on multi-formal Laurent series

Dai¹,

Wang²,

Ye³

2006

Acta Arith.

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1. Introduction. Continued fractions [8,14] are a useful tool in many number theoretical problems and in numerical computing. It is well known that the simple continued fraction expansion of a single real number gives the best solution to its rational approximation problem. Many people have contrived to construct multidimensional continued fractions in dealing with the rational approximation problem for multi-reals. One construction is the Jacobi-Perron algorithm (JPA) (see [1]). This algorithm and its modifications have been extensively studied [6,7,10,13]. These algorithms are adapted to study the same problem for multi-formal Laurent series [2,4,11,12]. But none of them guarantees the best rational approximation in general. In this paper, we deal with the multi-rational approximation problem over the formal Laurent series field F ((z −1 )): given an element r ∈ F ((z −1 )) m , find p ∈ F [z] m and q ∈ F [z] such that p/q approximates r as close as possible while deg(q) is bounded.We propose a new continued fraction algorithm for multi-formal Laurent series. It is proved that this algorithm guarantees best rational approximations for multi-formal Laurent series.The paper is organized as follows: Section 2 deals with the indexed valuation of F ((z −1 )) m . Section 3 contains the detailed definition of the problem of optimal rational approximation of multi-formal Laurent series. Section 4 proposes an algorithm called multidimensional continued fraction algorithm (m-CFA, for short), which produces a multi-continued fraction expansion C(r) for any given multi-formal Laurent series r. Section 5 shows that C(r) satisfies three basic conditions. Section 6 states the main results of this pa-

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Diophantine Approximation and Continued Fraction Expansions of Algebraic Power Series in Positive Characteristic

Cited by 19 publications

References 8 publications

On a two-valued sequence and related continued fractions in power series fields

On a two-valued sequence and related continued fractions in power series fields

Hyperquadratic continued fractions and automatic sequences

Multi-continued fraction algorithm on multi-formal Laurent series

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