Asymmetric stability for the Brunn–Minkowski-type inequality

Prove a stability theorem for two measurable subsets A,B ⊂ G of a compact Lie group G such that µ_G(AB)^{1/(d_G−d_H)} ≤ (1 + ε) ( µ_G(A)^{1/(d_G−d_H)} + µ_G(B)^{1/(d_G−d_H)} ), establishing that A and B must be close in structure to neighbourhoods of a proper closed subgroup achieving equality.

Background

The authors establish a symmetric stability result (A ≈ neighbourhood of a maximal subgroup) for near-minimal doubling but point out that an asymmetric two-set version is open. Such a result would mirror quantitative stability for Brunn–Minkowski in Euclidean spaces but faces additional non-commutative challenges.