We prove that a uniformly bounded system of orthonormal functions satisfying
the ψ2 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 ψ2 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 ψ2 condition is
Sidon.Comment: 22 pages, no figures. v2: minor edits based on referee comments v3:
further very minor edit