# Measure growth of small sets in $$\mathrm{SO}(3,\mathbb{R})$$

21.03.2023 15:00 - 17:00

Tran Chieu Minh (NU Singapore)

Let $$\mathrm{SO}(3,\mathbb{R})$$ be the 3D-rotation group equipped with the real-manifold topology and the normalized Haar measure $$\mu$$. Confirming a conjecture by Breuillard and Green, we show that if $$A \subseteq \mathrm{SO}(3,\mathbb{R})$$ is open and has sufficiently small measure, then $$\mu(A^2) > 3.99 \mu(A)$$. We also show a more general result for the product of two sets, which can be seen as a Brunn-Minkowski-type inequality for sets with small measure in $$\mathrm{SO}(3,\mathbb{R})$$. (Joint with Yifan Jing and Ruixiang Zhang)

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