Sidonicity and variants of Kaczmarz’s problem

Bourgain, Jean; Lewko, Mark

Annales de l'Institut Fourier, Tome 67 (2017) no. 3, pp. 1321-1352.

On démontre les propriétés suivantes pour un système de fonctions orthogonales, uniformement bornées et satisfaisant la condition ${\psi }_2$ : (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 ${\psi }_2$ 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 ${\psi }_2$ est de Sidon.

We prove that a uniformly bounded system of orthonormal functions satisfying the ${\psi }_2$ 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 ${\psi }_2$ 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 ${\psi }_2$ condition is Sidon.

Classification : 43A46, 42C05
Keywords: Sidon, Fourier series, orthogonal system, lacunary functions
Mot clés : Sidon, Série de Fourier, Systéme orthogonal, fonction lacunaires

[1] Bednorz, Witold; Latała, Rafał On the boundedness of Bernoulli processes, Ann. Math., Volume 180 (2014) no. 3, pp. 1167-1203 | DOI

[2] Bourgain, Jean Sidon sets and Riesz products, Ann. Inst. Fourier, Volume 15 (1985) no. 1, pp. 137-148 | DOI

[3] Bourgain, Jean On the distribution of polynomials on high-dimensional convex sets, Geometric aspects of functional analysis, Proc. Isr. Semin., GAFA, Isr. (1989–90) (Lecture Notes in Mathematics), Volume 1469, Springer, 1991, pp. 127-137

[4] Elton, John Sign-embeddings of $l_n^1$, Trans. Am. Math. Soc., Volume 279 (1983) no. 1, pp. 113-124

[5] Graham, Colin C.; Hare, Kathryn E. Interpolation and Sidon sets for compact groups, CMS Books in Mathematics, Springer, 2013, xvii+249 pages

[6] Lewko, Mark Allison; Lewko Orthonormal systems in linear spans, Anal. PDE, Volume 7 (2014) no. 1, pp. 97-115 | DOI

[7] Lopez, Jorge M.; Ross, Kenneth A. Sidon Sets, Lecture Notes in Pure and Applied Mathematics, 13, Marcel Dekker, 1975, v+193 pages

[8] Marcus, Michael B.; Pisier, Gilles Random Fourier series with applications to harmonic analysis, Annals of Mathematics Studies, 101, Princeton University Press and University of Tokyo Press, 1981, v+150 pages

[9] The Scottish Book. Mathematics from the Scottish Café (Mauldin, R.Daniel, ed.), Birkhäuser, 1981, xiii+268 pages (Including selected papers presented at the Scottish Book Conference held at North Texas State University, Denton, Tex., May 1979)

[10] Pajor, Alain Plongement de $l_k^1$ complexe dans les espaces de Banach, Séminaire de géométrie des espaces de Banach, Paris VII, 1983. Tomes I, II (Publications Mathématiques de l?Université Paris VII), Volume 18, Université Paris VII, 1984, pp. 139-148

[11] Pisier, Gilles On uniformly bounded orthonormal Sidon systems (https://arxiv.org/abs/1602.02430)

[12] Pisier, Gilles Ensembles de Sidon et processus gaussiens, C. R. Acad. Sci., Paris, Sér. A, Volume 286 (1978), pp. 671-674

[13] Preston, Christopher Banach spaces arising from some integral inequalities, Math. J., Indiana Univ., Volume 20 (1971), pp. 997-1015 | DOI

[14] Rider, Daniel Randomly continuous functions and Sidon sets, Duke Math. J., Volume 42 (1975) no. 4, pp. 759-764 | DOI

[15] Rudin, Walter Some theorems on Fourier coefficients, Proc. Am. Math. Soc., Volume 10 (1959), pp. 855-859 | DOI

[16] Rudin, Walter Trigonometric series with gaps, J. Math. Mech., Volume 9 (1960), pp. 203-227

[17] Spencer, Joel Six standard deviations suffice, Trans. Am. Math. Soc., Volume 289 (1985) no. 2, pp. 679-706 | DOI

[18] Talagrand, Michel Regularity of Gaussian processes, Acta Math., Volume 159 (1987) no. 1-2, pp. 99-149 | DOI

[19] Talagrand, Michel Majorizing measures without measures, Ann. Probab., Volume 29 (2001) no. 1, pp. 411-417 | DOI

[20] Talagrand, Michel Upper and lower bounds for stochastic processes. Modern methods and classical problems, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3, 60, Springer, 2014, xv+626 pages