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ON A PROBLEM POSED BY MAHLER

Part of: Diophantine approximation, transcendental number theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  11 November 2015

DIEGO MARQUES  and
JOHANNES SCHLEISCHITZ
DIEGO MARQUES
Affiliation:
Department of Mathematics, University of Brasilia, Brasilia, Df, Brazil email diego@mat.unb.br
JOHANNES SCHLEISCHITZ*
Affiliation:
Institute of Mathematics, University of Natural Resources and Life Sciences, Vienna, Augasse 2–6, 1090 Vienna, Austria email johannes.schleischitz@boku.ac.at
*
✉johannes.schleischitz@boku.ac.at
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Abstract

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Maillet proved that the set of Liouville numbers is preserved under rational functions with rational coefficients. Based on this result, a problem posed by Mahler is to investigate whether there exist entire transcendental functions with this property or not. For large parametrized classes of Liouville numbers, we construct such functions and moreover we show that they can be constructed such that all their derivatives share this property. We use a completely different approach than in a recent paper, where functions with a different invariant subclass of Liouville numbers were constructed (though with no information on derivatives). More generally, we study the image of Liouville numbers under analytic functions, with particular attention to $f(z)=z^{q}$, where $q$ is a rational number.

Keywords

Liouville numberstranscendental functionexceptional setcontinued fractions

MSC classification

Primary: 11J04: Homogeneous approximation to one number
Secondary: 11J82: Measures of irrationality and of transcendence 11K60: Diophantine approximation
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 100 , Issue 1 , February 2016 , pp. 86 - 107
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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