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

We analyze the local bahaviour 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

H-s(K boolean AND B(x, r)) <= (2r)(s) (1)

and

(2r)(s) <= P-s(K boolean AND B(x, r)) (2)

for all x is an element of K and all 0 < r < r(0), where H-s denotes the s-dimensional Hausdorff measure and 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 H-s(K) and the exact value of the s-dimensional packing measure P-s(K) of self-similar sets K.