Self-similar sets with optimal coverings and packings

Abstract

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.

Advantages of the Hausdorff centered measure from the computability point of view

“…In [14] it is proved that under strong separation condition (SSC) these optimal values are attained. We say that the self-similar set E satisfies the strong separation condition (SSC) if the union E = n i=1 f i (E) is disjoint.…”

“…We say that the self-similar set E satisfies the strong separation condition (SSC) if the union E = n i=1 f i (E) is disjoint. The ideas in [23] and [14] together with Theorem 3 gives readily the value of the Hausdorff centered measure. Remark 6.…”

Hausdorff measure of uniform self-similar fractals

“…Let x, y ∈ E such that diam(E) = x − y . If both x and y lies in E Γ , then diam(E) ≤ diam(E Γ ), which contradicts our assumption (12). Hence, w.l.o.g.…”

“…[1,12,15]) introduced a characterization of H D (E) in terms of an inverse density. In this paper we rely on the following results.…”

Asymptotic order of quantization for Cantor distributions in terms of Euler characteristic, Hausdorff and Packing measure

“…Recently (cf. [24]) it was also shown for higher-dimensional self-similar fractals, satisfying the strong separation condition, that the Hausdorff measure equals the inverse of the maximal density of the fractal and the Packing measure equals the inverse of the minimal density of the fractal. In case of N = 2 and d = 1, the non-convergence of the sequence (n r D V n,r (μ)) n∈N , i.e.…”