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We show that if $\partial\mathcal{R}$ is the boundary of the range ofsuper-Brownian motion and dim denotes Hausdorff dimension, then withprobability one, for any open set $U$, $\partial\mathcal{R}\cap U\neq\emptyset$implies $$\text{dim}(U\cap\partial\mathcal{R})=\begin{cases}4-2\sqrt2\approx1.17&\text{ if }d=2\\ \frac{9-\sqrt{17}}{2}\approx 2.44&\text{if }d=3. \end{cases}$$ This improves recent results of the last two authors(arxiv:1711.03486) by working with the actual topological boundary, rather thanthe boundary of the zero set of the local time, and establishing a local resultfor the dimension.

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