## Nombres transcendants et ensembles normaux.(French)Zbl 0177.07203

A set $$E$$ of real numbers is a normal set if there exists a sequence $$\Lambda = (\lambda_n)$$ such that $$(x\lambda_n)$$ is uniformly distributed $$\pmod 1$$ if and only if $$x$$ is an element of $$E$$. It is proved in this article that the set of real transcendental numbers is an intersection of a countable family of normal sets.
Since the publication of this result, Y. Meyer [Acta Arith. 16, 347–350 (1970; Zbl 0213.32802)], J. F. Colombeau [C. R. Acad. Sci., Paris, Sér. A 269, 270–272 (1969; Zbl 0177.07301)] and F. Dress [J. Number Theory 2, 352–362 (1970; Zbl 0199.37202)] have improved the result and simplified the demonstration. The method of Colombeau and Dress is the following. They first prove that if $$B_1, B_2,\ldots$$ are normal sets, then $$\cap_{s=1}^\infty B_s$$ is a normal set. Noticing that $$\mathbb R - \alpha\mathbb Q$$ $$(\alpha\in\mathbb R)$$ is a normal set, they conclude that $$\cap_{s=1}^\infty (\mathbb R - \alpha_s\mathbb Q)$$ is a normal set. By allowing the $$\alpha_s$$’s to be real algebraic numbers, one thus proves that the set of real transcendental numbers is a normal set.

### MSC:

 11K06 General theory of distribution modulo $$1$$ 11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.

### Citations:

Zbl 0213.32802; Zbl 0177.07301; Zbl 0199.37202
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