On the Centered Hausdorff Measure of the Sierpinski Gasket

Irregularity Index and Spherical Densities of the Penta-Sierpinski Gasket

“…Proof. Parts (i) and (ii) are proved in Lemma 4 of [11], and Parts (iii), (iv), (v) and (vi) are proved in Lemma 10 of [14]. Let B(x, d) with x ∈ P 02 , and 0 < d ≤ δ.…”

“…Proof. This lemma was proved in Lemma 8 of [11] for subsets and collections of k-cylinders of the Sierpinski gasket. Notice that for i = i 1 i 2 i 3 ...i k ∈ M k , and since…”

“…3.2 and 3.3), which are the natural extensions of Lebesgue measures to general metric spaces, and which we call metric measures. In order to illustrate the incipient state of knowledge in this respect, we mention that the only connected self-similar set with a non-integer dimension for which the spectra of asymptotic densities of some metric measure are known is the Sierpinski gasket S, for which α(S) and Spec(α, S) for α ∈ {C s , P s } were computed in [10], [11] and [14], respectively.…”

“…If a self-similar set E is α-exact, then α(E) can be, in the terminology of [11], C-computable for α (i.e., continuous computable), meaning that the task of computation of α(E) can be reduced to the solution of an optimisation problem of a continuous function in a compact domain. Thus, the computation of α(E) is, at least theoretically, amenable.…”

“…If this task can be accomplished, then we say that E is A-computable for α (i.e., algorithmic-computable [11]). See in [7] and [8] that a totally disconnected self-similar set E is an example of a C-computable and A-computable set for α ∈ {C s , P s }, and see [8] for the issue of the rate of convergence of the estimations of α(E) in that case.…”