Proceedings of the Japan Academy, Series A, Mathematical Sciences

On transcendental analytic functions mapping an uncountable class of $U$-numbers into Liouville numbers

Diego Marques and Josimar Ramirez

Full-text: Open access


In this paper, we shall prove, for any $m\geq 1$, the existence of an uncountable subset of $U$-numbers of type $\leq m$ (which we called the set of \textit{$m$-ultra numbers}) for which there exists uncountably many transcendental analytic functions mapping it into Liouville numbers.

Article information

Proc. Japan Acad. Ser. A Math. Sci., Volume 91, Number 2 (2015), 25-28.

First available in Project Euclid: 2 February 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11Jxx: Diophantine approximation, transcendental number theory [See also 11K60]

Transcendental functions $m$-ultra number Liouville height


Marques, Diego; Ramirez, Josimar. On transcendental analytic functions mapping an uncountable class of $U$-numbers into Liouville numbers. Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), no. 2, 25--28. doi:10.3792/pjaa.91.25.

Export citation


  • P. Erdös, Representations of real numbers as sums and products of Liouville numbers, Michigan Math. J. 9 (1962), 59–60.
  • K. Mahler, Some suggestions for further research, Bull. Austral. Math. Soc. 29 (1984), no. 1, 101–108.
  • D. Marques and C. G. Moreira, On a variant of a question proposed by K. Mahler concerning Liouville numbers, Bull. Austral. Math. Soc. 91 (2015), no. 1, 29–33.
  • D. Marques, On the arithmetic nature of hypertranscendental functions at complex points, Expo. Math. 29 (2011), no. 3, 361–370.
  • A. J. Van der Poorten, Transcendental entire functions mapping every algebraic number field into itself, J. Austral. Math. Soc. 8 (1968), 192–193.
  • P. Stäckel, Ueber arithmetische Eingenschaften analytischer Functionen, Math. Ann. 46 (1895), no. 4, 513–520.
  • P. Stäckel, Arithmetische Eigenschaften Analytischer Functionen, Acta Math. 25 (1902), no. 1, 371–383.
  • M. Waldschmidt, Algebraic values of analytic functions, J. Comput. Appl. Math. 160 (2003), no. 1-2, 323–333.
  • M. Waldschmidt, Diophantine approximation on linear algebraic groups, Grundlehren der Mathematischen Wissenschaften, 326, Springer, Berlin, 2000.