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

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]).

Are there transcendental entire functions f(z) such that if [xi] is any Liouville number, then so is f([xi])?

Then the number [[xi].sub.A] := [0, [a.sub.1], [a.sub.2], [a.sub.3], [a.sub.4], ...] is an ultra-strong Liouville number. In fact, if [0, [a.sub.1], ..., [a.sub.n]] = [p.sub.n]/[q.sub.n], then, by construction, [mathematical expression not reproducible] (here, we used the well-known inequality ([a.sub.1] + 1) ...

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 [9] that the sum of two strong Liouville numbers is a Liouville number).

A real number [xi] is called a Liouville number, if there exist a sequence of rational numbers [([p.sub.k]/[q.sub.k]).sub.k], with [q.sub.k] > 1, and a sequence of positive real numbers [([w.sub.k]).sub.k] such that [limsup.sub.k[right arrow][infinity]][w.sub.k] = +[infinity] and

The most classical example of a Liouville number is the Liouville's constant' (see [1]), 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.

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.