Submission date: 25 January 2024

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 every ε > 0 and for all measurable subsets A of small enough Haar measure, we have
μ_G(A^2) > (2^{d_G - d_H}-ε)μ_G(A)
where d_H is the maximal dimension of a proper closed subgroup H. This settles a conjecture of Breuillard and Green and recovers - with completely different methods - a recent result of Jing--Tran--Zhang corresponding to the case G=SO_3(ℝ).
Going beyond the scope of this conjecture, our methods enable us to prove a stability result asserting that the only subsets close to saturating this inequality are essentially neighbourhoods of proper subgroups i.e. of the form H_δ :={g ∈ G: d(g,H)<δ} where H denotes a maximal closed subgroup, d denotes a bi-invariant distance on G and δ > 0.
Our approach relies on two a priori unrelated toolsets: optimal transports and its recent applications to the Brunn--Minkowski inequality, and the structure theory of compact approximate subgroups.

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Full text arXiv 2401.14062: pdf, ps.