Measure doubling for small sets in SO(3,R)

Confirming a conjecture by Breuillard and Green, we show that for every epsilon>0 there is a delta>0 such that if A is an open subset of SO(3,R) with measure at most delta, then the measure of A^2 is strictly greater than (4-epsilon) times the measure of A. This is joint work with Chieu-Minh Tran and Ruixiang Zhang.