Effective dimension of points visited by Brownian motion

We consider the individual points on a Martin-Löf random path of Brownian motion. We show that (1) Khintchine’s law of the iterated logarithm holds at almost all points; and (2) there exist points (besides the trivial example of the origin) having effective dimension <1. The proof of (1) shows that, for almost all times t, the path f is Martin-Löf random relative to t and so the effective dimension of (t,f(t)) is 2.