Minimal doubling for small subsets in compact Lie groups

Abstract

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 $$\hat{I}{¼}_G(A^2)\ge \left(2^{d_G-d_H}-C\hat{I}{¼}_G(A{)}^{\frac{2}{d_G-d_H}}\right)\hat{I}{¼}_G(A)$$ 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 $\hat{I}\mu >0$ and for any pair of sufficiently small measurable subsets $A,B$ a Brunn--Minkowski-type inequality holds: $$\hat{I}{¼}_G(AB{)}^{\frac{1}{d_G-d_H}}\ge (1-\hat{I}\mu )\left(\hat{I}{¼}_G(A{)}^{\frac{1}{d_G-d_H}}+\hat{I}{¼}_G(B{)}^{\frac{1}{d_G-d_H}}\right).$$ 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 $$H_{\hat{I}}´:=\{g\in G:d(g,H)<\hat{I}´\}$$ where $H$ denotes a proper closed subgroup of maximal dimension, $d$ denotes a bi-invariant distance on $G$ and $\hat{I}´>0$. 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.

44 pages. New proof of Proposition 1.6 relies on discretization/multi-scale analysis of approximate groups, rather than on an unpublished result due to Carolino. The proof is now, in principle, quantitative. Submitted