Effective dimension of points visited by Brownian motion

We consider the individual points on a Martin-Löf 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-Löf random relative to $t$ and so the effective dimension of $(t,f(t))$ is 2.

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