Irregularity index and asymptotic spectrum of spherical densities of the Penta-Sierpinski gasket

We compute the centred Hausdorff measure, $C^{s}(\mathbf{P})\sim2.44$, and the packing measure, $P^{s}(\mathbf{P})\sim6.77$, of the penta-Sierpinski gasket, $\mathbf{P}$, with explicit error bounds. We also compute the full spectra of asymptotic spherical densities of these measures in $\mathbf{P}$, which, in contrast with that of the Sierpinski gasket consists of a unique interval. These results allow us to compute the irregularity index of $\mathbf{P}$, $\mathcal{I}(\mathbf{P})\sim0.6398$, which we define for any self-similar set $E$ with open set condition as $\mathcal{I}(E)=1-\frac{C^{s}(E))}{P^{s}(E)}$. Mathematics Subject Classification. 28A78, 28A80, 28A75.