Higher order rectifiability of measures via averaged discrete curvatures

We provide a sufficient geometric condition for $\mathbb R^n$ to be countably $(\mu,m)$ rectifiable of class $\mathcal C^{1,\alpha}$ (using the terminology of Federer), where $\mu$ is a Radon measure having positive lower density and finite upper density $\mu$ almost everywhere. Our condition involves integrals of certain many-point interaction functions (discrete curvatures) which measure flatness of simplexes spanned by the parameters.