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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2401.14062 |
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| _version_ | 1866909208672206848 |
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| author | Machado, Simon |
| author_facet | Machado, Simon |
| contents | 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 $$μ_G(A^2) \geq \left(2^{d_G-d_H} - Cμ_G(A)^{\frac{2}{d_G-d_H}}\right)μ_G(A)$$ 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 $G=SO_3(\mathbb{R})$. As is often the case, the above doubling inequality stems from a special case of general product-set estimates. We prove that for all $ε>0$ and for any pair of sufficiently small measurable subsets $A,B$ a Brunn--Minkowski-type inequality holds:
$$ μ_G(AB)^{\frac{1}{d_G-d_H}} \geq (1-ε)\left( μ_G(A)^{\frac{1}{d_G-d_H}} + μ_G(B)^{\frac{1}{d_G-d_H}}\right).$$
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 $$H_δ:=\{g \in G: d(g,H)<δ\}$$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_14062 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimal doubling for small subsets in compact Lie groups Machado, Simon Group Theory Combinatorics 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 $$μ_G(A^2) \geq \left(2^{d_G-d_H} - Cμ_G(A)^{\frac{2}{d_G-d_H}}\right)μ_G(A)$$ 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 $G=SO_3(\mathbb{R})$. As is often the case, the above doubling inequality stems from a special case of general product-set estimates. We prove that for all $ε>0$ and for any pair of sufficiently small measurable subsets $A,B$ a Brunn--Minkowski-type inequality holds: $$ μ_G(AB)^{\frac{1}{d_G-d_H}} \geq (1-ε)\left( μ_G(A)^{\frac{1}{d_G-d_H}} + μ_G(B)^{\frac{1}{d_G-d_H}}\right).$$ 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 $$H_δ:=\{g \in G: d(g,H)<δ\}$$ 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. |
| title | Minimal doubling for small subsets in compact Lie groups |
| topic | Group Theory Combinatorics |
| url | https://arxiv.org/abs/2401.14062 |