Restrictions of Brownian motion

It is classical that the zero set and the set of record times of a linear Brownian motion have Hausdorff dimension \(1/2\) almost surely. Can we find a larger random subset on which a Brownian motion is monotone? Perhaps surprisingly, the answer is negative. We outline the short proof, which is an application of Kaufman's dimension doubling theorem for planar Brownian motion. This is a joint work with Yuval Peres.