  We provide a sufficient geometric condition for $\mathbb{R}^n$ to be
countably $(\mu,m)$ rectifiable of class $\mathscr{C}^{1,\alpha}$ (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 simplices spanned by the parameters.
