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A NOTE ON LACUNARY POWER SERIES WITH RATIONAL COEFFICIENTS

Part of: Series expansions Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  11 November 2015

DIEGO MARQUES ,
JOSIMAR RAMIREZ  and
ELAINE SILVA
DIEGO MARQUES*
Affiliation:
Departamento de Matemática, Universidade de Brasília, Brasília, DF, Brazil email [email protected]
JOSIMAR RAMIREZ
Affiliation:
Departamento de Matemática, Universidade de Brasília, Brasília, DF, Brazil email [email protected]
ELAINE SILVA
Affiliation:
Departamento de Matemática, Universidade de Brasília, Brasília, DF, Brazil email [email protected]
*
✉[email protected]
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Abstract

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In this note, we prove that for any ${\it\nu}>0$, there is no lacunary entire function $f(z)\in \mathbb{Q}[[z]]$ such that $f(\mathbb{Q})\subseteq \mathbb{Q}$ and $\text{den}f(p/q)\ll q^{{\it\nu}}$, for all sufficiently large $q$.

Keywords

Liouville numberslacunary power series

MSC classification

Primary: 11J82: Measures of irrationality and of transcendence
Secondary: 30B10: Power series (including lacunary series)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 3 , June 2016 , pp. 372 - 374
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Mahler, K., ‘Arithmetic properties of lacunary power series with integral coefficients’, J. Aust. Math. Soc. 5 (1965), 56–64.Google Scholar
Mahler, K., ‘Some suggestions for further research’, Bull. Aust. Math. Soc. 29 (1984), 101–108.Google Scholar
Maillet, E., Introduction à la Théorie des Nombres Transcendants et des Propriétés Arithmétiques des Fonctions (Gauthier-Villars, Paris, 1906).Google Scholar
Marques, D. and Moreira, C. G., ‘A variant of a question proposed by K. Mahler concerning Liouville numbers’, Bull. Aust. Math. Soc. 91 (2015), 29–33.Google Scholar
Marques, D. and Ramirez, J., ‘On transcendental analytic functions mapping an uncountable class of U-numbers into Liouville numbers’, Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), 25–28.CrossRefGoogle Scholar
Marques, D. and Schleischitz, J., ‘On a problem posed by Mahler’, J. Aust. Math. Soc., to appear.Google Scholar