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Liouville numbers, Liouville sets and Liouville fields


Authors: K. Senthil Kumar, R. Thangadurai and M. Waldschmidt
Journal: Proc. Amer. Math. Soc. 143 (2015), 3215-3229
MSC (2010): Primary 11J82
DOI: https://doi.org/10.1090/proc/12408
Published electronically: April 22, 2015
MathSciNet review: 3348766
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Abstract | References | Similar Articles | Additional Information

Abstract: Following earlier work by É. Maillet 100 years ago, we introduce the definition of a Liouville set, which extends the definition of a Liouville number. We also define a Liouville field, which is a field generated by a Liouville set. Any Liouville number belongs to a Liouville set $ \mathsf {S}$ having the power of continuum and such that $ \mathbf {Q}\cup \mathsf {S}$ is a Liouville field.


References [Enhancements On Off] (What's this?)

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  • [2] J. Liouville, Sur des classes très étendues de quantités dont la valeur n'est ni algébrique, ni même réductible des irrationnelles algébriques, J. Math. Pures et Appl. 18 (1844) 883-885, and 910-911.
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Additional Information

K. Senthil Kumar
Affiliation: Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad, 211019, India
Address at time of publication: The Institute of Mathematical Sciences, 4th Cross Road, CIT Campus, Taramani, Chennai, 600113, India
Email: senthilkk@imsc.res.in

R. Thangadurai
Affiliation: Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad, 211019, India
Email: thanga@hri.res.in

M. Waldschmidt
Affiliation: Institut de Mathématiques de Jussieu, Théorie des Nombres Case courrier 247, Université Pierre et Marie Curie (Paris 6), Paris Cedex 05, France
Email: miw@math.jussieu.fr

DOI: https://doi.org/10.1090/proc/12408
Keywords: Liouville number, Liouville set, Liouville field, continuum, $G_\delta$-subsets
Received by editor(s): May 27, 2013
Published electronically: April 22, 2015
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2015 American Mathematical Society

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