Geometric criteria for $C^{1,α}$ rectifiability

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

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