Diophantine approximation and continued fraction expansion for quartic power series over $$\pmb {\mathbb {F}}_{3}$$

2021 ◽  
Author(s):  
Khalil Ayadi ◽  
Awatef Azaza ◽  
Salah Beldi
Keyword(s):  
Power Series ◽  
Diophantine Approximation ◽  
Continued Fraction ◽  
Continued Fraction Expansion

scholarly journals Diophantine approximation by continued fractions

1991 ◽  
Vol 51 (2) ◽  
pp. 324-330 ◽  
Author(s):  
Jingcheng Tong
Keyword(s):  
Diophantine Approximation ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Irrational Number ◽  
Continued Fraction Expansion ◽  
Image Position

AbstractLet ξ be an irrational number with simple continued fraction expansion be its ith convergent. Let Mi = [ai+1,…, a1]+ [0; ai+2, ai+3,…]. In this paper we prove that Mn−1 < r and Mn R imply which generalizes a previous result of the author.

scholarly journals The continuous Diophantine approximation mapping of Szekeres

1995 ◽  
Vol 59 (2) ◽  
pp. 148-172 ◽  
Author(s):  
Jeffrey C. Lagarias ◽  
Andrew D. Pollington
Keyword(s):  
Diophantine Approximation ◽  
Continued Fraction ◽  
Approximation Properties ◽  
Continued Fraction Expansion ◽  
Open Problems ◽  
Unit Square ◽  
Alternative Characterization ◽  
Continuous Analogue

AbstractSzekeres defined a continuous analogue of the additive ordinary continued fraction expansion, which iterates a map T on a domain which can be identified with the unit square [0, 1]2. Associated to it are continuous analogues of the Lagrange and Markoff spectrum. Our main result is that these are identical with the usual Lagrange and Markoff spectra, respectively; thus providing an alternative characterization of them.Szekeres also described a multi-dimensional analogue of T, which iterates a map Td on a higherdimensional domain; he proposed using it to bound d-dimensional Diophantine approximation constants. We formulate several open problems concerning the Diophantine approximation properties of the map Td.

scholarly journals Continued fractions and diophantine equations in positive characteristic

2021 ◽  
Vol 109 (123) ◽  
pp. 143-151
Author(s):  
Khalil Ayadi ◽  
Awatef Azaza ◽  
Salah Beldi
Keyword(s):  
Power Series ◽  
Finite Field ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Diophantine Equations ◽  
Positive Characteristic ◽  
Continued Fraction Expansion ◽  
Ring Of Polynomials ◽  
Algebraic Power Series

We exhibit explicitly the continued fraction expansion of some algebraic power series over a finite field. We also discuss some Diophantine equations on the ring of polynomials, which are intimately related to these power series.

scholarly journals Some inequalities for diophantine approximation by continued fractions

1990 ◽  
Vol 41 (2) ◽  
pp. 249-253
Author(s):  
Jingcheng Tong
Keyword(s):  
Diophantine Approximation ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Irrational Number ◽  
Continued Fraction Expansion ◽  
General Method

Let ξ be an irrational number with simple continued fraction expansion ξ = [a0;a1,a2,…,an,…], let pn/qn be its nth convergent and let θn = qn|qnξ − pn|. In this paper a general method is introduced to deduce a series of inequalities involving the triple (θn−1, θn, θn+1).

The Smallest Pisot Element in the Field of Formal Power Series Over a Finite Field

2013 ◽  
Vol 56 (2) ◽  
pp. 258-264
Author(s):  
A. Chandoul ◽  
M. Jellali ◽  
M. Mkaouar
Keyword(s):  
Power Series ◽  
Finite Field ◽  
Formal Power Series ◽  
Continued Fraction ◽  
Laurent Series ◽  
Minimal Polynomial ◽  
Ordered Set ◽  
Pisot Number ◽  
Continued Fraction Expansion ◽  
Formal Power

Abstract.Dufresnoy and Pisot characterized the smallest Pisot number of degree n ≥ 3 by giving explicitly its minimal polynomial. In this paper, we translate Dufresnoy and Pisot’s result to the Laurent series case. The aim of this paper is to prove that the minimal polynomial of the smallest Pisot element (SPE) of degree n in the field of formal power series over a finite field is given by P(Y) = Yn–XYn-1–αn where α is the least element of the finite field 픽q\{0} (as a finite total ordered set). We prove that the sequence of SPEs of degree n is decreasing and converges to αX: Finally, we show how to obtain explicit continued fraction expansion of the smallest Pisot element over a finite field.

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