Geometric criteria for C1,α$C^{1,\alpha }$‐rectifiability

We prove criteria for $\mathcal{H}^k$-rectifiability of subsets $\mathbb{R}^n$ with $C^{1,\alpha}$ maps, $0<\alpha\leq 1$, in terms suitable approximate tangent paraboloids. also provide a version the case when there is not an priori plane, measuring on dyadic scales how close set to lying $k$-plane. then discuss relation similar involving Peter Jones' $\beta$ numbers, particular proving that sufficient condition boundedness small $r$ $r^{-\alpha}\beta_p(x,r)$ $\mathcal{H}^k$-a.e. $x$ and any $1\leq p\leq \infty$.