Sidonicity and variants of Kaczmarz’s problem
Annales de l'Institut Fourier, Volume 67 (2017) no. 3, p. 1321-1352

We prove that a uniformly bounded system of orthonormal functions satisfying the ψ 2 condition: (1) must contain a Sidon subsystem of proportional size, (2) must satisfy the Rademacher–Sidon property, and (3) must have its five-fold tensor satisfy the Sidon property. On the other hand, we construct a uniformly bounded orthonormal system that satisfies the ψ 2 condition but which is not Sidon. These problems are variants of Kaczmarz’s Scottish book problem (problem 130) which, in its original formulation, was answered negatively by Rudin. A corollary of our argument is a new elementary proof of Pisier’s theorem that a set of characters satisfying the ψ 2 condition is Sidon.

On démontre les propriétés suivantes pour un système de fonctions orthogonales, uniformement bornées et satisfaisant la condition ψ 2  : (1) le système contient un sous-ensemble de Sidon de taille proportionnelle, (2) il satisfait la propriété Rademacher–Sidon et (3) le produit tensoriel d’ordre cinq a la propriété de Sidon. Par contre, on construit un exemple d’un tel système ψ 2 montrant que le système lui-même n’est pas nécessairement de Sidon. Il s’agit de variantes du problème de Kaczmarz (probleme 130 dans le “Scottish book”) qui, dans sa formulation initiale, fut résolu négativement par Rudin. Comme corollaire, on obtient une nouvelle démonstration élémentaire d’un théorème de Pisier montrant qu’un système de caractères satisfaisant la propriété ψ 2 est de Sidon.

Received : 2015-06-04
Revised : 2016-07-15
Accepted : 2016-09-15
Published online : 2017-05-31
Classification:  43A46,  42C05
Keywords: Sidon, Fourier series, orthogonal system, lacunary functions
     author = {Bourgain, Jean and Lewko, Mark},
     title = {Sidonicity and variants of Kaczmarz's problem},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {3},
     year = {2017},
     pages = {1321-1352},
     doi = {10.5802/aif.3111},
     language = {en},
     url = {}
Bourgain, Jean; Lewko, Mark. Sidonicity and variants of Kaczmarz’s problem. Annales de l'Institut Fourier, Volume 67 (2017) no. 3, pp. 1321-1352. doi : 10.5802/aif.3111.

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