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

Abstract

Let $\mathrm{SO}(3,\mathbb{R})$ be the 3D-rotation group equipped with the real-manifold topology and the normalized Haar measure $\mu$. Resolving a problem by Breuillard and Green, we show that if $A\subseteq \mathrm{SO}(3,\mathbb{R})$ is an open subset with 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 $\mathrm{SO}(3,\mathbb{R})$.

40 pages. Comments are welcome