Spectral Gap Properties of the Unitary Groups: Around Rider’s Results on Non-commutative Sidon Sets

Abstract

We present a proof of Rider’s unpublished result that the union of two Sidon sets in the dual of a non-commutative compact group is Sidon, and that randomly Sidon sets are Sidon. Most likely this proof is essentially the one announced by Rider and communicated in a letter to the author around 1979 (lost by him since then). The key fact is a spectral gap property with respect to certain representations of the unitary groups U(n) that holds uniformly over n. The proof crucially uses Weyl’s character formulae. We survey the results that we obtained 30 years ago using Rider’s unpublished results. Using a recent different approach valid for certain orthonormal systems of matrix-valued functions, we give a new proof of the spectral gap property that is required to show that the union of two Sidon sets is Sidon. The latter proof yields a rather good quantitative estimate. Several related results are discussed with possible applications to random matrix theory.