Rectifiability of  Pointwise Doubling Measures in Hilbert Space

Abstract: Jones’ beta numbers measure the flatness of a set at various scales and windows.  Since their introduction, beta numbers have served as an important tool to relate the geometric structure of sets and measures and to measure-theoretic quantities.  We will extend results of Badger and Schul to show that an L^2 variant of the beta numbers can be used to characterize rectifiable pointwise doubling measures in Hilbert space.  We will also discuss results for the related notions of graph rectifiability and fractional rectifiability.