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On the Arithmetic Behavior of Liouville Numbers Under Rational Maps

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Abstract

In 1972, Alniaçik proved that every strong Liouville number is mapped into the set of \(U_m\)-numbers, for any non-constant rational function with coefficients belonging to an m-degree number field. In this paper, we generalize this result by providing a larger class of Liouville numbers (which, in particular, contains the strong Liouville numbers) with this same property (this set is sharp is a certain sense).

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Acknowledgements

We would like to thank the reviewer for his/her positive and insightful comments on the manuscript. Part of the preparation of this paper was made during a visit of A.P.C. and D.M. to IMPA during its Summer Program 2019. They thank this institution for its support, hospitality and excellent working conditions. A.P.C. was supported in part by CNPq Universal 01/2016-427722/2016-0 grant. D.M. is supported by a Productivity and Research scholarship from CNPq-Brazil. P.T. has been supported by Specific Research Project of Faculty of Science, University of Hradec Králové, No. 2101, 2018.

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Authors and Affiliations

  1. Instituto de Matemática e Estatística, Universidade Federal de Goiás, Goiânia, Brazil

    Ana Paula Chaves

  2. Departamento de Matemática, Universidade de Brasília, Brasília, Brazil

    Diego Marques

  3. Faculty of Science, University of Hradec Králové, Hradec Králové, Czech Republic

    Pavel Trojovský

Authors
  1. Ana Paula Chaves
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  2. Diego Marques
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  3. Pavel Trojovský
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Correspondence to Diego Marques.

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Chaves, A.P., Marques, D. & Trojovský, P. On the Arithmetic Behavior of Liouville Numbers Under Rational Maps. Bull Braz Math Soc, New Series 52, 803–813 (2021). https://doi.org/10.1007/s00574-020-00232-7

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  • DOI: https://doi.org/10.1007/s00574-020-00232-7

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