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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.
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