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Abstract: We present a proof of Rider's unpublished result that the union of two Sidon sets in the dual of a non-commutative compact group is Sidon, and that randomly Sidon sets are Sidon.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 We survey the results that we obtained 30 years ago using Rider's unpublished results.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.