Given a measurable set $K\subset \mathbb{R}^d$ we consider the occupation ratio $$f_r(x)=vol(K\cap B(x,r))/r^d$$ and especially the asymptotics when $r\to 0$. When $K$ has a fractal boundary and $x$ is on the boundary, it is not clear whether most $x$'s have a large occupation ratio.

Has anyone already seen this quantity? Is it studied somewhere?

If you want a precise question, let $K$ be a non-negligible set with self-similar boundary with dimension $s$ and positive lower Minkowski content. Do we have $$\liminf_r\int_{\partial K}f_r(x)d\nu(x)>0$$ where $\nu$ is the $s$-dimensional Hausdorff measure? For the Von Koch flake i think the answer is yes but I don't see a general scheme.

EDIT: I insist on the fact that the set $K$ itself is not fractal, but its boundary is.

Real and Complex Analysis, chapter 8) tells you that $\lim f_r = 1_K$ almost everywhere which, unfortunately, does not say anything about the integral w.r.t. the Hausdorff measure on $\partial K$. $\endgroup$ – Jochen Wengenroth Aug 20 '14 at 13:36