The Breuillard–Green conjecture for minimal doubling in compact Lie groups

Let $G$ be a compact Lie group of dimension $d_G$, and let $d_H$ be the maximal dimension of a proper closed subgroup $H$.

Breuillard–Green conjecture. The minimal value of the doubling constant for subsets of small measure should be about

$$2^{d_G-d_H}.$$

This generalizes the conjecture that small subsets of $\operatorname{SO}_3(\mathbb{R})$ have doubling at least $4-\epsilon$. The paper proves a sharp asymptotic lower bound, settling this conjecture.