This is an announcement for the paper "Sidonicity and variants of Kaczmarz's problem" by Jean Bourgain and Mark Lewko.

Abstract: 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 5-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.

Archive classification: math.CA math.FA math.PR

Remarks: 22 pages, no figures

Submitted from: mlewko@gmail.com

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1504.05290

or

http://arXiv.org/abs/1504.05290