Sidonicity and variants of Kaczmarz's problem

TL;DR

This paper investigates the properties of orthonormal systems satisfying the  condition, establishing their relation to Sidon systems, and provides new insights and counterexamples related to Kaczmarz's problem and Pisier's theorem.

Contribution

It proves that  systems with boundedness conditions contain Sidon subsystems and satisfy the Rademacher-Sidon property, while also constructing counterexamples and offering a new proof of Pisier's theorem.

Findings

 systems contain Sidon subsystems of proportional size

Such systems satisfy the Rademacher-Sidon property

Counterexample of  system not being Sidon

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.