Before revision: -- one that satisfies uniform bounds of the form for some c>0, all x in the support of \mu, and all sufficiently small r . Our main result is that if x_1,\ldots  x_n are sampled from a d-Ahlfors regular measure on \mathbb{R and E_\alpha\left(x_1,\ldots,x_n\right) denotes the \alpha-weight of the minimal spanning tree on x_1,\ldots,x_n:
After revision: -- one that satisfies uniform bounds of the form prove that the fractal dimension of a metric space equipped with an Ahlfors regular measure can be recovered from the persistent homology of random samples . Our main result is that if x_1,\ldots x_n are sampled from a d-Ahlfors regular measure on \mathbb{R and E_\alpha\left(x_1,\ldots,x_n\right) denotes the \alpha-weight of the minimal spanning tree on x_1,\ldots,x_n: