On the packing measure of the Sierpinski gasket

Abstract

We show that the s-dimensional packing measure P s (S) of the Sierpinski gasket S; where s = log 3 log 2 is the similarity dimension of S; satis…es 1:6677 P s (S) 1:6713. The formula presented in Theorem 6 enables the achievement of the above measure bounds for this non-totally disconnected set as it shows that the symmetries of the Sierpinski gasket can be exploited to simplify the density characterization of P s obtained in [33] for self-similar sets satisfying the so-called Open Set Condition. Thanks to the reduction obtained in Theorem 6 we are able to handle the problem of computability of P s (S) with a suitable algorithm.

Irregularity Index and Spherical Densities of the Penta-Sierpinski Gasket

“…The algorithms for the estimation of P s (P) and C s (P) are based on parts (i) and (ii) of Theorem 6.3 respectively. They are similar to those described in [10] and [11] for the estimation of the packing and centred Hausdorff measures of the Sierpinski gasket, respectively. The modifications required for the adaptation to the penta-Sierpinski gasket are obvious using Theorem 6.3.…”

“…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.…”