Density theorems for Hausdorff and packing measures of self-similar sets

Abstract

We analyze the local behaviour of the Hausdorff measure and the packing measure of self-similar sets. In particular, if K is a self-similar set whose Hausdorff dimension and packing dimension equal s, a special case of our main results says that if K satisfies the Open Set Condition, then there exists a number r 0 such that 1 $${\mathcal{H}}^{s}(K \cap B(x, r)) \leq (2r)^{s}$$ and 2 $$(2r)^{s} \leq {\mathcal{P}}^{s}(K \cap B(x, r))$$ for all x ∈ K and all 0 < r < r 0, where $${\mathcal{H}}^{s}$$ denotes the s-dimensional Hausdorff measure and $${\mathcal{P}}^{s}$$ denotes the s-dimensional packing measure. Inequality (1) and inequality (2) are used to obtain a number of very precise density theorems for Hausdorff and packing measures of self-similar sets. These density theorems can be applied to compute the exact value of the s-dimensional Hausdorff measure $${\mathcal{H}}^{s}(K)$$ and the exact value of the s-dimensional packing measure $${\mathcal{P}}^{s}(K)$$ of self-similar sets K.