Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 

 

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
Published electronically: April 22, 2015
MathSciNet review: 3348766
Full-text PDF

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?)

  • [1] P. Erdős, Representations of real numbers as sums and products of Liouville numbers, Michigan Math. J. 9 (1962), 59–60. MR 0133300
  • [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.
  • [3] É. Maillet, Introduction la théorie des nombres transcendants et des propriétés arithmétiques des fonctions, Paris, 1906.
  • [4] E. Maillet, Sur quelques propriétés des nombres transcendants de Liouville, Bull. Soc. Math. France 50 (1922), 74–99 (French). MR 1504810

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11J82

Retrieve articles in all journals with MSC (2010): 11J82


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