Quantitative Reifenberg theorem for measures
Nick Edelen, Aaron Naber, Daniele Valtorta
TL;DR
The paper develops a quantitative Reifenberg theory for general measures in $\mathbb{R}^n$ using Jones' $\beta$-numbers, proving effective mass and packing bounds away from a $k$-rectifiable core under weak $\beta$-number control and without density assumptions. It introduces a corona-type decomposition into good and bad regions, constructs approximating bi-Lipschitz manifolds via a stopping-time tree, and derives both local (packing) and global (mass/rectifiability) conclusions, with stronger results when density bounds are available. The framework yields discrete and quantitative Reifenberg-type results for sets and measures, and connects to upper Ahlfors-regularity and rectifiability through a careful balance of mass concentration and geometric tilting. The approach hinges on a generalized center of mass, a good/bad ball dichotomy, and a two-tier tree argument that propagates control across scales while accounting for holes and excess sets. Collectively, the results extend Reifenberg-type regularity to a broad class of measures, with applications to discrete measures, sets, and Calderón–Zygmund theory under $\beta$-bounds.
Abstract
We study generalizations of Reifenberg's Theorem for measures in $\mathbb R^n$ under assumptions on the Jones' $β$-numbers, which appropriately measure how close the support is to being contained in a subspace. Our main results, which holds for general measures without density assumptions, give effective measure bounds on $μ$ away from a closed $k$-rectifiable set with bounded Hausdorff measure. We show examples to see the sharpness of our results. Under further density assumptions one can translate this into a global measure bound and $k$-rectifiable structure for $μ$. Applications include quantitative Reifenberg theorems on sets and discrete measures, as well as upper Ahlfor's regularity estimates on measures which satisfy $β$-number estimates on all scales.