  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unpublished result that randomly Sidon sets are Sidon, which implies that the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.
