We 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 i.i.d. ~ samples from a d-Ahlfors regular measure on a metric space, and E  _\alpha\left(x_1,\ldots,x_n\right) denotes the \alpha-weight of the minimum spanning tree on x_1,\ldots,x_n: %DIFDELCMD < \[E_\alpha\left(x_1,\ldots,x_n\right)=e\in T\left(x_1,\ldots,x_n\right) |e|^\alpha\,,\] %%% then%DIFDELCMD < \[E_\alpha\left(x_1,\ldots,x_n\right)\approx n^{d-\alpha{d}}\] %%% \[E_\alpha^0\left(x_1,\ldots,x_n\right)=e\in T\left(x_1,\ldots,x_n\right) |e|^\alpha\,,\] \[C_1\leq n^{-d-\alpha{d}} E^0_\alpha\left(x_1,\ldots,x_n\right)\leq C_2\,\] with high probability as n\rightarrow \infty. In particular, %DIFDELCMD < \[\log\big(E_\alpha(x_1,\ldots,x_n)\big)/\log(n)\longrightarrow (d-\alpha)/d\,.\] %%% \[\log\big(E^0_\alpha(x_1,\ldots,x_n)\big)/\log(n)\longrightarrow (d-\alpha)/d\,.\] 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)  . We also prove analogous results for weighted sums defined in terms of higher dimensional persistent homology.
We 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 i.i.d. samples from a d-Ahlfors regular measure on a metric space, and E ^0 _\alpha\left(x_1,\ldots,x_n\right) denotes the \alpha-weight of the minimum spanning tree on x_1,\ldots,x_n: %DIFDELCMD < \[E_\alpha\left(x_1,\ldots,x_n\right)=e\in T\left(x_1,\ldots,x_n\right) |e|^\alpha\,,\] %%% %DIFDELCMD < \[E_\alpha\left(x_1,\ldots,x_n\right)\approx n^{d-\alpha{d}}\] %%% \[E_\alpha^0\left(x_1,\ldots,x_n\right)=e\in T\left(x_1,\ldots,x_n\right) |e|^\alpha\,,\] then there exist constants 0<C_1\leq C_2 so that\[C_1\leq n^{-d-\alpha{d}} E^0_\alpha\left(x_1,\ldots,x_n\right)\leq C_2\,\] with high probability as n\rightarrow \infty. In particular, %DIFDELCMD < \[\log\big(E_\alpha(x_1,\ldots,x_n)\big)/\log(n)\longrightarrow (d-\alpha)/d\,.\] %%% \[\log\big(E^0_\alpha(x_1,\ldots,x_n)\big)/\log(n)\longrightarrow (d-\alpha)/d\,.\] This is a generalization of a result of Steele (1988) from the non-singular 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(x_1,\ldots,x_n\right) does not exist with high probability . We also prove analogous results for weighted sums defined in terms of higher dimensional persistent homology.
D
~
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^0
D
then
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then there exist constants 0<C_1\leq C_2 so that
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absolutely continuous
non-singular
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. Our result is best possible, in the sense that there exist Ahlfors regular measures for which the limit \lim_{n\rightarrow\infty
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x_1,\ldots,x_n
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does not exist with high probability