  We present a proof of Rider's unpublished result that the union of two Sidon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obtained 30 years ago using Rider's unpublished results.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.
