Asymmetric stability of the Brunn--Minkowski inequality in compact Lie groups

We show a stability result for the recently established Brunn--Minkowski inequality in compact simple Lie groups. Namely, we prove that if two compact subsets $A, B$ of a compact simple Lie group $G$ satisfy $$\mu(AB)^{1/d'} \leq (1 + \epsilon)\left(\mu(A)^{1/d'} + \mu(B)^{1/d'}\right)$$ where $AB$ is the Minkowski product $\{ab : a \in A, b \in B\}$, $d'$ denotes the minimal codimension of a proper closed subgroup and $\mu$ is a Haar measure, then $A$ and $B$ must approximately look like neighbourhoods of a proper subgroup $H$ of codimension $d'$, with an error that depends quantitatively on $d', \epsilon$ and the ratio $\frac{\mu(A)}{\mu(B)}$. This result implies an improved error rate in the Brunn--Minkowski inequality in compact simple Lie groups $$\mu(AB)^{\frac{1}{d'}} \geq (1-C\mu(A)^{\frac{2}{d'}})\left(\mu(A)^{\frac{1}{d'}} + \mu(B)^{\frac{1}{d'}}\right)$$ sharp, up to the constant $C$ which depends on $d'$ and $\frac{\mu(A)}{\mu(B)}$ alone. Our approach builds upon an earlier paper of the author proving the Brunn--Minkowski inequality, and stability in the case $A=B$. We employ a combinatorial multi-scale analysis and study so-called density functions. Additionally, the asymmetry between $A$ and $B$ introduces new challenges, requiring the use of non-abelian Fourier theory and stability results for the Pr\'ekopa--Leindler inequality.