Period-doubling Continued Fractions are Algebraic in Characteristic 2
Hu, Yining1; Lasjaunias, Alain2
1 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan (P. R. China)
2 Mathematical Institute Bordeaux University, Bordeaux (France)
Annales de l'Institut Fourier, Online first, 9 p.

For an arbitrary pair of distinct and non constant polynomials, a and b in 𝔽 2 [t], we build a continued fraction in 𝔽 2 ((1/t)) whose partial quotients are only equal to a or b. In a previous work of the first author and Han, the authors considered two cases where the sequence of partial quotients represents in each case a famous and basic 2-automatic sequence, both defined in a similar way by morphisms. They could prove the algebraicity of the corresponding continued fractions for several pairs (a,b) in the first case (the Prouhet–Thue–Morse sequence) and gave the proof for a particular pair for the second case (the period-doubling sequence). Recently Bugeaud and Han proved the algebraicity for an arbitrary pair in the first case. Here we give a short proof for an arbitrary pair in the second case.

Pour une paire de polynômes non constants et distincts a et b dans 𝔽 2 [t], on construit une fraction continue dans 𝔽 2 ((1/t)) dont les quotients partiels sont égales à a ou b. Dans un travail précédent de la première auteure et Han, les auteurs ont considéré deux cas où les quotients partiels forment des suites 2-automatiques bien connues, définies de façon similaire comme point fixe des morphismes. Ils ont pu démontré l’algébricité des fractions continues associées pour plusieures paires (a,b) dans le premier cas (la suite de Prouhet–Thue–Morse) et ont donné la preuve pour une paire dans le deuxième cas (la suite de doublement de période). Récemment, Bugeaud et Han ont démontré l’algébricité pour une paire arbitraire dans le premier cas. Ici, on donne une preuve courte pour une paire arbitraire dans le deuxième cas.

Received:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3614
Classification: 11B85, 11J70, 11B50, 11Y65, 05A15, 11T55
Keywords: Algebraicity, Automatic sequence, Continued fraction, Period-doubling sequence.
Mot clés : Algébricité, suite automatique, fraction continue, suite de doublement de période.

Hu, Yining 1; Lasjaunias, Alain 2

1 School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan (P. R. China)
2 Mathematical Institute Bordeaux University, Bordeaux (France) 
@unpublished{AIF_0__0_0_A75_0,
     author = {Hu, Yining and Lasjaunias, Alain},
     title = {Period-doubling {Continued} {Fractions} are {Algebraic} in {Characteristic} 2},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2024},
     doi = {10.5802/aif.3614},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - Hu, Yining
AU  - Lasjaunias, Alain
TI  - Period-doubling Continued Fractions are Algebraic in Characteristic 2
JO  - Annales de l'Institut Fourier
PY  - 2024
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3614
LA  - en
ID  - AIF_0__0_0_A75_0
ER  - 
%0 Unpublished Work
%A Hu, Yining
%A Lasjaunias, Alain
%T Period-doubling Continued Fractions are Algebraic in Characteristic 2
%J Annales de l'Institut Fourier
%D 2024
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3614
%G en
%F AIF_0__0_0_A75_0
Hu, Yining; Lasjaunias, Alain. Period-doubling Continued Fractions are Algebraic in Characteristic 2. Annales de l'Institut Fourier, Online first, 9 p.

[1] Allouche, Jean-Paul; Shallit, Jeffrey Automatic sequences. Theory, Applications, Generalizations, Cambridge University Press, 2003, xvi+571 pages | DOI | MR | Zbl

[2] Baum, Leonard E.; Sweet, Melvin M. Badly approximable power series in characteristic 2, Ann. Math., Volume 105 (1977) no. 3, pp. 573-580 | DOI | MR | Zbl

[3] Bugeaud, Yann; Han, Guo-Niu The Thue–Morse continued fractions in characteristic 2 are algebraic (2023) (to be published in Acta Arithmetica)

[4] Hu, Yining; Han, Guo-Niu On the algebraicity of Thue–Morse and period-doubling continued fractions, Acta Arith., Volume 203 (2022) no. 4, pp. 353-381 | DOI | MR | Zbl

[5] Lasjaunias, Alain A survey of Diophantine approximation in fields of power series, Monatsh. Math., Volume 130 (2000) no. 3, pp. 211-229 | DOI | MR | Zbl

[6] Lasjaunias, Alain Continued fractions (2017) (https://arxiv.org/abs/1711.11276)

Cited by Sources: