Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-27T20:14:33.316Z Has data issue: false hasContentIssue false

ON A VARIANT OF A QUESTION PROPOSED BY K. MAHLER CONCERNING LIOUVILLE NUMBERS

Published online by Cambridge University Press:  09 July 2014

DIEGO MARQUES*
Affiliation:
Departamento de Matemática, Universidade de Brasília, Brasília, DF, Brazil email [email protected]
CARLOS GUSTAVO MOREIRA
Affiliation:
Instituto de Matemática Pura e Aplicada, Rio de Janeiro, RJ, Brazil email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this note we shall prove the existence of an uncountable subset of Liouville numbers (which we call the set of ultra-Liouville numbers) for which there exist uncountably many transcendental analytic functions mapping the subset into itself.

Type
Research Article
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

Bugeaud, Y., Approximation by Algebraic Numbers, Cambridge Tracts in Mathematics, 160 (Cambridge University Press, New York, 2004).CrossRefGoogle Scholar
Chaves, A. P. and Marques, D., ‘An explicit family of U m-numbers’, Elem. Math. 69 (2014), 1822.CrossRefGoogle Scholar
Erdős, P., ‘Representations of real numbers as sums and products of Liouville numbers’, Michigan Math. J. 9 (1962), 5960.Google Scholar
Liouville, J., ‘Sur des classes très-étendues de quantités dont la valeur n’est ni algébrique, ni même reductible à des irrationnelles algébriques’, C. R. Acad. Sci. Paris 18 (1844), 883885.Google Scholar
Mahler, K., ‘Some suggestions for further research’, Bull. Aust. Math. Soc. 29 (1984), 101108.Google Scholar
Maillet, E., Introduction à la Théorie des Nombres Transcendants et des Propriétés Arithmétiques des Fonctions (Gauthier-Villars, Paris, 1906).Google Scholar
Stäckel, P., ‘Ueber arithmetische Eingenschaften analytischer Functionen’, Math. Ann. 46(4) (1895), 513520.Google Scholar
Waldschmidt, M., ‘Algebraic values of analytic functions’, Proc. Int. Conf. on Special Functions and their Applications, Chennai, 2002, J. Comput. Appl. Math. 160(1–2) (2003), 323333.Google Scholar