On the most visited sites of planar Brownian motion

Let (B-t : t >= 0) be a planar Brownian motion and define a family of gauge functions phi(alpha)(s) = log (1/s) (alpha) for alpha > 0. If alpha < 1 we show that almost surely there exists a point x in the plane such that H-phi alpha ({t >= 0 : B-t = x }) > 0, but if alpha > 1 almost surely H-phi alpha ({t >= 0 : B-t = x}) = 0 simultaneously for all x is an element of R-2. This resolves a longstanding open problem posed by S.J. Taylor in 1986.