Before revision:  In particular,  \[\log\big(E^0_\alpha(x_1,\ldots,x_n)\big)/\log(n)\longrightarrow (d-\alpha)/d\,.\] This is a generalization of a result of Steele (1988) from the absolutely continuous case to the fractal setting  n^{-d-\alpha{d}} E^0_\alpha}\left(\right)  .
After revision:  In particular,  \[\log\big(E^0_\alpha(x_1,\ldots,x_n)\big)/\log(n)\longrightarrow (d-\alpha)/d\,.\] This is a generalization of a result of Steele (1988) from the absolutely continuous case to the fractal setting . Our result is best possible, in the sense that there exist Ahlfors regular measures for which the limit \lim_{n\rightarrow\infty n^{-d-\alpha{d}} E^0_\alpha}\left(\right) .