On the spectrum of irrationality exponents of Mahler numbers

Dzmitry Badziahin

doi:10.5802/jtnb.1090

Dzmitry Badziahin ¹

¹ School of Mathematics and Statistics University of Sydney NSW 2006, Australia

Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 2, pp. 431-453.

Abstract
Résumé

We consider Mahler functions $f (z)$ which satisfy the functional equation $f (z) = \frac{A (z)}{B (z)} f (z^{d})$ where $\frac{A (z)}{B (z)}$ is in $ℚ (z)$ and $d \geq 2$ is an integer. We prove that, for any integer $b$ with $| b | \geq 2$ , 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 $f (z) = \frac{A (z)}{B (z)} f (z^{d})$ , où $\frac{A (z)}{B (z)}$ est dans $ℚ (z)$ et $d \geq 2$ est un entier. Nous montrons que, pour tout entier $b$ vérifiant $| b | \geq 2$ , 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: 2018-11-12
Accepted: 2019-06-14
Published online: 2019-10-29

Zbl

DOI: 10.5802/jtnb.1090

Classification: 11J82, 05A15, 11B85
Keywords: Mahler functions, Mahler Numbers, Irrationality exponent, Hankel determinant

Author's affiliations:

Dzmitry Badziahin ¹

¹ School of Mathematics and Statistics University of Sydney NSW 2006, Australia

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {On the spectrum of irrationality exponents of {Mahler} numbers},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {431--453},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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     doi = {10.5802/jtnb.1090},
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Dzmitry Badziahin. 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/articles/10.5802/jtnb.1090/

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