A real number [xi] is called a Liouville number, if there exist infinitely many rational numbers [([p.
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]).
Are there transcendental entire functions f(z) such that if [xi] is any Liouville number, then so is f([xi])?
Here, we shall define an uncountable subclass of the strong Liouville numbers which we named as ultra-strong Liouville numbers: a real number [xi] is called an ultra-strong Liouville number, if the sequence [([p.
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).
The most classical example of a Liouville number is the Liouville's constant' (see ), defined as a decimal with a 1 in each decimal place corresponding to n
In particular, an irrational number is a Liouville number if and only if it is not Diophantine.
The question which motivated this note is to see "how many" replacements of 0's by 1's one can make between two consecutive 1's in the decimal expansion of l, in order to guarantee that this new number is still a Liouville number.
1/9 - l which is a Liouville number (since the difference between a Liouville number and a rational number is always a Liouville number).
On variations of the Liouville constant which are also Liouville numbers
Diego MARQUES and Carlos Gustavo MOREIRA Communicated by Masaki KASHIWARA, M.
A note on transcendental entire functions mapping uncountable many Liouville numbers
into the set of Liouville numbers
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.