On the spectrum of irrationality exponents of Mahler numbers
Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 2, pp. 431-453.

We consider Mahler functions f(z) which satisfy the functional equation f(z)=A(z) B(z)f(z d ) where A(z) B(z) is in (z) and d2 is an integer. We prove that, for any integer b with |b|2, either f(b) is rational or its irrationality exponent is rational. We also compute the exact value of the irrationality exponent of f(b) as soon as the continued fraction expansion of the Mahler function f(z) is known. This improves the result of Bugeaud, Han, Wen, and Yao [6] where only an upper bound of the irrationality exponent was provided.

Nous considérons les fonctions de Mahler f(z) qui véri-fient l’équation fonctionnelle f(z)=A(z) B(z)f(z d ), où A(z) B(z) est dans (z) et d2 est un entier. Nous montrons que, pour tout entier b vérifiant |b|2, ou bien f(b) est rationnel, ou bien son exposant d’irrationalité est rationnel. En outre, nous déterminons la valeur exacte de l’exposant d’irrationalité de f(b) lorsque l’on connaît le développement en fraction continue de la fonction de Mahler f(z). Cela améliore un résultat de Bugeaud, Han, Wen et Yao [6], qui ne donne qu’une borne supérieure de cet exposant.

Received : 2018-11-13
Accepted : 2019-06-15
Published online : 2019-10-29
DOI : https://doi.org/10.5802/jtnb.1090
Classification:  11J82,  05A15,  11B85
Keywords: Mahler functions, Mahler Numbers, Irrationality exponent, Hankel determinant
@article{JTNB_2019__31_2_431_0,
     author = {Dzmitry Badziahin},
     title = {On the spectrum of irrationality exponents of Mahler numbers},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {2},
     year = {2019},
     pages = {431-453},
     doi = {10.5802/jtnb.1090},
     language = {en},
     url={jtnb.centre-mersenne.org/item/JTNB_2019__31_2_431_0/}
}
Badziahin, Dzmitry. On the spectrum of irrationality exponents of Mahler numbers. Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 2, pp. 431-453. doi : 10.5802/jtnb.1090. https://jtnb.centre-mersenne.org/item/JTNB_2019__31_2_431_0/

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