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Sidonicity and variants of Kaczmarz’s problem

2017; Association of the Annals of the Fourier Institute; Volume: 67; Issue: 3 Linguagem: Inglês

10.5802/aif.3111

1777-5310

Jean Bourgain, Mark Lewko,

Mathematical Approximation and Integration

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.