2024/01/25 by Simon Machado, Machado, Simon · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.2401.14062
openalex publication_date 2024/01/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 dG, we show that for for all measurable subsets A, we have μG(A2) ≥ (2dG-dH - CμG(A)(2)/(dG-dH))μG(A) where dH 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=SO3(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 ε>0 and for any pair of sufficiently small measurable subsets A,B a Brunn--Minkowski-type inequality holds: μG(AB)(1)/(dG-dH) ≥ (1-ε)( μG(A)(1)/(dG-dH) + μG(B)(1)/(dG-dH)). 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 Hδ:=\g ∈ G: d(g,H)lt;δ\ where H denotes a proper closed subgroup of maximal dimension, d denotes a bi-invariant distance on G and δ>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.