On uniformly bounded orthonormal Sidon systems

TL;DR

This paper proves that uniformly bounded orthonormal systems are 2-fold tensor Sidon, improving previous bounds, and explores conditions under which such systems are randomly Sidon, with applications to systems like Peter-Weyl decompositions.

Contribution

It establishes sharper bounds on tensor Sidon properties of orthonormal systems and characterizes when they are randomly Sidon, extending results to non-Abelian group systems.

Findings

Any uniformly bounded ψ₂(C)-orthonormal system is 2-fold tensor Sidon.

A uniformly bounded orthonormal system is randomly Sidon iff it is 4-fold tensor Sidon.

Union of two Sidon sets is Sidon.

Abstract

In answer to a question raised recently by Bourgain and Lewko, we show, with their paper's terminology, that any uniformly bounded $\psi_2 (C)$-orthonormal system ($\psi_2 (C)$ is a variant of subGaussian)is 2-fold tensor Sidon. This sharpens their result that it is 5-fold tensor Sidon. The proof is somewhat reminiscent of the author's original one for (Abelian) group characters, based on ideas due to Drury and Rider. However, we use Talagrand's majorizing measure theorem in place of Fernique's metric entropy lower bound. We also show that a uniformly bounded orthonormal system is randomly Sidon iff it is 4-fold tensor Sidon, or equivalently $k$-fold tensor Sidon for some (or all) $k\ge 4$. Various generalizations are presented, including the case of random matrices, for systems analogous to the Peter-Weyl decomposition for compact non-Abelian groups. In the latter setting we also include a new proof of Rider's unpublished result that randomly Sidon sets are Sidon, which implies that the union of two Sidon sets is Sidon.