On uniformly bounded orthonormal Sidon systems

Gilles Pisier

Published 2016-02-07Version 1

In answer to a question raised recently by Bourgain and Lewko, we show, with their paper's terminology, that any $\psi_2 (C)$-orthonormal system ($\psi_2 (C)$ is a variant of subGaussian) is two-fold tensor Sidon (which improves their result that it is five-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. 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 In the latter setting we also include a new proof of Rider's unpublished result that randomly Sidon sets are Sidon, which implies that the union of two Sidon sets is Sidon.