Let G be a connected unimodular group equipped with a (left and hence right) Haar measure μ, and suppose A, B ⊆ G are nonempty and compact. An inequality by Kemperman gives us
μ(AB) ≥ min {μ(A)+μ(B), μ(G) }.
We obtain characterizations of G, A, and B such that the equality holds. This is a Vosper’s theorem in the continuous non-abelian setting, and provides a complete answer to a question asked by Kemperman in 1964. We also get near equality versions of the above results with uniform linear bound for connected compact groups. This confirms conjectures made by Griesmer and by Tao and can be seen as a Freiman (3k-4)-theorem up to a constant factor for this setting. Our result can be applied to obtain the first measure expansion gap result for connected compact simple Lie groups.
This is joint work with Chieu-Minh Tran.
This video is part of the Webinar in Additive Combinatorics series, and this is their YouTube channel.
