Structure of exact minimisers of the doubling ratio at fixed measure

Characterize the structure of measurable subsets A ⊂ G that minimise the ratio µ_G(A^2)/µ_G(A) among all measurable A with µ_G(A) = α, for a given connected Lie group G and parameter α > 0.

Background

The authors note that, despite recent advances, exact minimisers for the doubling ratio are largely unknown in non-commutative settings. Their main results and stability theorem suggest neighbourhoods of maximal subgroups are strong candidates, but a complete structural description of minimisers remains open. They highlight special interest in concrete groups and mention known non-existence in certain nilpotent cases.

References

Problem 8.3. Let G be a connected Lie group and α > 0. What is the structure of a minimiser of the ratio µG(A )/µG(A) for A ranging through measurable subsets with µG(A) = α? Concrete groups G of particular interest are S3 (R), S2 (R). Note however that such minimizers never exist in the Heisenberg group H 3R) [42].

— Minimal doubling for small subsets in compact Lie groups (2401.14062 - Machado, 25 Jan 2024) in Problem 8.3, Section 8.2