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ON EXCEPTIONAL SETS: THE SOLUTION OF A PROBLEM POSED BY K. MAHLER

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  12 May 2016

DIEGO MARQUES  and
JOSIMAR RAMIREZ
DIEGO MARQUES*
Affiliation:
Departamento de Matemática, Universidade de Brasília, Brasília, 70910-900, Brazil email [email protected]
JOSIMAR RAMIREZ
Affiliation:
Departamento de Matemática, Universidade de Brasília, Brasília, 70910-900, Brazil email [email protected]
*
✉[email protected]
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Abstract

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In this paper, we shall prove that any subset of $\overline{\mathbb{Q}}$, which is closed under complex conjugation, is the exceptional set of uncountably many transcendental entire functions with rational coefficients. This solves an old question proposed by Mahler [Lectures on Transcendental Numbers, Lecture Notes in Mathematics, 546 (Springer, Berlin, 1976)].

Keywords

exceptional settranscendental function

MSC classification

Primary: 11J81: Transcendence (general theory)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 94 , Issue 1 , August 2016 , pp. 15 - 19
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Huang, J., Marques, D. and Mereb, M., ‘Algebraic values of transcendental functions at algebraic points’, Bull. Aust. Math. Soc. 82 (2010), 322–327.CrossRefGoogle Scholar
Mahler, K., ‘Arithmetic properties of lacunary power series with integral coefficients’, J. Aust. Math. Soc. 5 (1965), 56–64.CrossRefGoogle Scholar
Mahler, K., Lectures on Transcendental Numbers, Lecture Notes in Mathematics, 546 (Springer, Berlin, 1976).CrossRefGoogle Scholar
Stäckel, P., ‘Ueber arithmetische Eingenschaften analytischer Functionen’, Math. Ann. 46 (1895), 513–520.CrossRefGoogle Scholar
Waldschmidt, M., ‘Algebraic values of analytic functions’, J. Comput. Appl. Math. 160 (2003), 323–333.CrossRefGoogle Scholar