It is well-known that the set of the Liouville numbers L is a [G.sub.[delta]]-dense set and therefore an uncountable set.
In his pioneering book, Maillet [5, Chapitre III] discusses some arithmetic properties of Liouville numbers. One of them is that, given a non-constant rational function f, with rational coefficients, if [xi] is a Liouville number, then so is f([xi]).
He also said that: "The difficulty of this problem lies of course in the fact that the set of all Liouville numbers is non-enumerable".
We remark about the existence of more specific classes of Liouville numbers in the literature, for example, the strong and semi-strong Liouville numbers (see, for instance, ).
We remark that the set L is different from all previously constructed sets in [6-8] since all those sets are [G.sub.[delta]]-dense while L does not have this property (in fact, it was proved in  that the sum of two strong Liouville numbers is a Liouville number).
In particular, there exist uncountable many transcendental entire functions taking the set of the ultra-strong Liouville numbers into the set of Liouville numbers.
Ramirez, On transcendental analytic functions mapping an uncountable class of U-numbers into Liouville numbers, Proc.
A real number [xi] is called a Liouville number, if there exist infinitely many rational numbers [([p.sub.n]/[q.sub.n]).sub.n[greater than or equal to]1], with [q.sub.n] [greater than or equal to] 1 and such that
Are there transcendental entire functions f(z) such that if [xi] is any Liouville number, then so is f([xi])?
On transcendental analytic functions mapping an uncountable class of U-numbers into Liouville numbers
Diego MARQUES and Josimar RAMIREZ Semisimple symmetric spaces without compact manifolds locally modelled thereon Yosuke MORITA Above two, communicated by Kenji FUKAYA, M.J.A.