Hausdorff-measure conjecture for persistent homology functionals

Let

be a $d$-Ahlfors regular measure on a metric space $M$, and let $\{x_n\}_{n\in\mathbb{N}}$ be i.i.d. samples from

. Write $E_\alpha^i(x_1,\ldots,x_n)$ for the persistent homology functional in degree $i$, and let $f$ denote the density of the absolutely continuous part of

with respect to the $d$-dimensional Hausdorff measure $\mathcal{H}^d$. **Hausdorff-measure conjecture.** If $\mathit{PH}_0(\operatorname{supp}\mu)$ is trivial and $0<\alpha<d$, then

\lim_{n\to\infty} n^{-(d-\alpha)/d}E_\alpha^0(x_1,\ldots,x_n)=c_0(\alpha,d)\int_M f(x)^{(d-\alpha)/d},d\mathcal{H}^d(x)

with probability one, where $c_0(\alpha,d)$ is continuous in $\alpha$ and $d$. Furthermore, if

is supported on $\mathbb{R}^m$, $d>\gamma_i^m$, and $\mathit{PH}_i(\operatorname{supp}\mu)$ is trivial, then

$$\lim_{n\to\infty} n^{-(d-\alpha)/d}E_\alpha^i(x_1,\ldots,x_n)=c_i(\alpha,d)\int_M f(x)^{(d-\alpha)/d}\,d\mathcal{H}^d(x)$$

with probability one, where $c_i(\alpha,d)$ is continuous in $\alpha$ and $d$.

This conjecture extends the corresponding asymptotic result from Lebesgue measure to Hausdorff measure under triviality of the persistent homology of the support; the source notes that it would exclude a counterexample to the general limit statement.