$k$-dimensional regularity classifications for $s$-fractals

Abstract:

We study subsets $E$ of ${{\mathbf {R}}^n}$ which are ${H^s}$ measurable and have $0 < {H^s}(E) < \infty$, where ${H^s}$ is the $s$-dimensional Hausdorff measure. Given an integer $k$, $s \leqslant k \leqslant n$, we consider six ($s$, $k$) regularity definitions for $E$ in terms of $k$-dimensional subspaces or surfaces of ${{\mathbf {R}}^n}$. If $s = k$, they all agree with the (${H^k}$, $k$) rectifiability in the sense of Federer, but in the case $s < k$ we show that only two of them are equivalent. We also study sets with positive lower density, and projection properties in connection with these regularity definitions.