Spectral Gap Properties of the Unitary Groups: Around Riders Results on Non-commutative Sidon Sets
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Springer International Publishing AG, part of Springer Nature 2018. We present a proof of Riders unpublished result that the union of two Sidon sets in the dual of a non-commutative compact group is Sidon, and that randomly Sidon sets are Sidon.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 Weyls character formulae. We survey the results that we obtained 30 years ago using Riders unpublished results.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.
Operator Theory: Advances and Applications, Vol. 261, 415--463, 2017, Springer
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