Minimal doubling for small subsets in compact Lie groups

We prove a sharp bound for the minimal doubling of a small measurable subset of a compact connected Lie group. Namely, let $G$ be a compact connected Lie group of dimension $d_G$, we show that for for all measurable subsets $A$, we have $${\mu }_G(A^2)\ge (2^{d_G-d_H}-C{\mu }_G(A{)}^{\frac{2}{d_G-d_H}}){\mu }_G(A)$$ where $d_H$ is the maximal dimension of a proper closed subgroup $H$ and $C>0$ is a dimensional constant. This settles a conjecture of Breuillard and Green, and recovers and improves - with completely different methods - a recent result of Jing–Tran–Zhang corresponding to the case $G=S{O}_3(\mathbb{R})$. As is often the case, the above doubling inequality stems from a special case of general product-set estimates. We prove that for all $\epsilon >0$ and for any pair of sufficiently small measurable subsets $A,B$ a Brunn–Minkowski-type inequality holds: $${\mu }_G(AB{)}^{\frac{1}{d_G-d_H}}\ge (1-\epsilon )({\mu }_G(A{)}^{\frac{1}{d_G-d_H}}+{\mu }_G(B{)}^{\frac{1}{d_G-d_H}})\mathrm{.}$$ Going beyond the scope of the Breuillard–Green conjecture, we prove a stability result asserting that the only subsets with close to minimal doubling are essentially neighbourhoods of proper subgroups i.e. of the form $$H_{\delta }:=\{g\in G:d(g,H)<\delta \}$$ where $H$ denotes a proper closed subgroup of maximal dimension, $d$ denotes a bi-invariant distance on $G$ and $\delta >0$. Our approach relies on a combination of two toolsets: optimal transports and its recent applications to the Brunn–Minkowski inequality, and the structure theory of compact approximate subgroups.