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Uniformly distributed sets and sets \(\Lambda(p)\). (Ensembles uniformément répartis et ensembles \(\Lambda(p)\).) (French) Zbl 1010.43007

Choquet, G. (ed.) et al., Séminaire d’initiation à l’analyse. 37ème et 38ème années: 1997/1999. Exposés 1 à 20. Paris: Univ. Pierre et Marie Curie, Publ. Math. Univ. Pierre Marie Curie. 121, 9 p. (2000).
This is the French version of the author’s paper [Ann. Inst. Fourier 49, 1853-1867 (1999; Zbl 0955.42009)]. In [Proc. Am. Math. Soc. 126, 3329-3333 (1998; Zbl 0907.43007)] the reviewer pointed out the existence of \(\Lambda(p)\)-sets \(\Lambda\subseteq N\) (i.e. \(L^1_\Lambda\) reflexive) which are not Rosenthal sets (i.e. \(L^\infty_\Lambda\not= {\mathcal C}_\Lambda\)): \({\mathcal C}_\Lambda\) contains subspaces isomorphic to \(c_0\); this follows essentially from two random constructions: the first one, stated without details by Y. Katznelson [Lect. Notes Math. 336, 148-152 (1973; Zbl 0274.42009)], to get \(\Lambda(p)\)-sets, and the second one, due to J. Bourgain [Isr. J. Math. 61, 39-72 (1988; Zbl 0642.28010)], to produce ergodic sequences \(\Lambda\) in \(N\) (which, by a result of F. Lust-Piquard [Semin. Géom. Espaces de Banach, Éc. Polytechn., Cent. Math. 1977/78, 26, 9p. (1978; Zbl 0386.46020)], implies that \({\mathcal C}_\Lambda\) contains a subspace isomorphic to \(c_0\)).
In the paper under review, the author generalizes Katznelson’s construction (and gives a complete proof of it); mainly, he gives a necessary and sufficient condition on some Littlewood-Paley partitions of \(N\) to get sets which are \(\Lambda(p)\)-sets for all \(p<+\infty\) (Théorème 17). He also generalizes Bourgain’s result, with a simpler proof, to get ergodic sequences in prescribed uniformly distributed sequences of integers. This allows him to get that the set of squares and the set of prime numbers contain sets which are uniformly distributed (in a broad sense) and which are \(\Lambda(p)\)-sets for all \(p<+\infty\) (Corollaire 8).
These results were strongly improved by the reviewer, H. Queffélec and L. Rodríguez-Piazza in [J. d’Analyse Math. 86, 105-138 (2002; Zbl 1018.43004)].
For the entire collection see [Zbl 0966.00021].

MSC:

43A46 Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
03E15 Descriptive set theory
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