Before revision:  \[\log\big(E_\alpha(x_1,\ldots,x_n)\big)/\log(n)\longrightarrow (d-\alpha)/d\,.\]  We also prove theorems about the asymptotic behavior of weighted sums defined in terms of higher-dimensional persistent homology.
After revision:  \[\log\big(E_\alpha(x_1,\ldots,x_n)\big)/\log(n)\longrightarrow (d-\alpha)/d\,.\] This is a generalization of a result of Steele (1988) from the absolutely continuous case to the fractal setting. We also prove theorems about the asymptotic behavior of weighted sums defined in terms of higher-dimensional persistent homology.