On the topological boundary of the range of super-Brownian motion

We show that if .R is the boundary of the range of super-Brownian motion and dim denotes Hausdorff dimension, then with probability one, for any open set U, U..R.. implies dim(U..R)={4.22...1.17if d=2,9.17.2.2.44if d=3. This improves recent results of the last two authors by working with the actual topological boundary, rather than the boundary of the zero set of the local time, and establishing a local result for the dimension.