On uniformly bounded orthonormal Sidon systems

In answer to a question raised recently by Bourgain and Lewko, we show that any uniformly bounded subGaussian orthonormal system is $\otimes^2$-Sidon. This sharpens their result that it is “5-fold tensor Sidon”, or $\otimes^5$-Sidon in our terminology. 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 if it is $\otimes^4$-tensor Sidon, or equivalently $\otimes^k$-Sidon for some (or all) $k \geq 4$. 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 In the latter setting we also include a new proof of Rider’s unpublished result that randomly Sidon sets are Sidon, which implies that the union of two Sidon sets is Sidon.

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Received 22 February 2016

Accepted 30 June 2016

Published 1 September 2017