Self-similar sets with optimal coverings and packings

We prove that if a self-similar set E in R-n with Hausdorff dimension s satisfies the strong separation condition, then the maximal values of the H-s-density on the class of arbitrary subsets of R-n and on the class of Euclidean balls are attained, and the inverses of these values give the exact values of the Hausdorff and spherical Hausdorff measure of E. We also show that a ball of minimal density exists, and the inverse density of this ball gives the exact packing measure of E. Lastly, we show that these elements of optimal densities allow us to construct an optimal almost covering of E by arbitrary subsets of R-n, an optimal almost covering of E by balls and an optimal packing of E. (c) 2007 Elsevier Inc. All rights reserved.