Effective dimension of points visited by Brownian motion

Abstract

We consider the individual points on a Martin-L\"of random path of Brownian motion. We show (1) that 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-L\"of random relative to $t$ and so the effective dimension of $(t,f(t))$ is 2.