C^ {1, α}  Reifenberg Theorems for Sets and Measures  

Abstract. We provide geometric sufficient conditions for Reifenberg flat sets with holes in Rⁿ to be parametrized by a C^1, α map. The conditions use a Jones type square function and all statements are quantitative in that the Hölder and Lipschitz constants of the parametrizations depend on such a function. We use these results to prove sufficient conditions for higher order rectifiability of sets and measures in Rⁿ. Key tools for the proof come from Guy David and Tatiana Toro’s parametrization of Reifenberg flat sets (with holes) in the Hölder and Lipschitz categories.