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

Summary.

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

$${\mathcal{H}}^{s}(K \cap B(x, r)) \leq (2r)^{s}$$

(1)

and

$$(2r)^{s} \leq {\mathcal{P}}^{s}(K \cap B(x, r))$$

(2)

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.