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 $$\mu (AB{)}^{1\mathrm{/}{d}^{\mathrm{\prime }}}\le (1+\epsilon )(\mu (A{)}^{1\mathrm{/}{d}^{\mathrm{\prime }}}+\mu (B{)}^{1\mathrm{/}{d}^{\mathrm{\prime }}})$$ where $AB$ is the Minkowski product $\{ab:a\in A,b\in B\}$, $d^{\mathrm{\prime }}$ 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 $d^{\mathrm{\prime }}$, with an error that depends quantitatively on $d^{\mathrm{\prime }},\epsilon$ 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 $$\mu (AB{)}^{\frac{1}{d^{\mathrm{\prime }}}}\ge (1-C\mu (A{)}^{\frac{2}{d^{\mathrm{\prime }}}})(\mu (A{)}^{\frac{1}{d^{\mathrm{\prime }}}}+\mu (B{)}^{\frac{1}{d^{\mathrm{\prime }}}})$$ sharp, up to the constant $C$ which depends on $d^{\mathrm{\prime }}$ 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 Prékopa–Leindler inequality.