Quantitative Reifenberg theorem for measures

This paper studies a quantitative version of Reifenberg-type theorems for measures in ${\mathbb{R}}^n$, replacing the classical uniform two-sided geometric flatness of sets by scale-by-scale quantitative flatness measured through Jones’ $\beta$-numbers. The central research question is: given a general nonnegative Borel-regular measure $\mu$ (with no density or Ahlfors regularity assumptions), how can one control the amount of $\mu$ that lies away from a $k$-dimensional rectifiable set, and how can one deduce rectifiability or mass bounds from quantitative $\beta$-number summability? This matters because many geometric-analytic problems (e.g., singular sets of PDEs, varifold regularity, and boundedness of Calderón–Zygmund operators on nonhomogeneous measures) require understanding the structure and size of measures whose supports may have holes, excess, and even lower-dimensional “concentrations,” none of which are allowed in classical Reifenberg theorems.

The authors’ key innovation is to work with $\beta$-numbers for measures rather than sets, and to allow only weak (measure-theoretic) control of $\beta$ at scales, rather than pointwise or uniform smallness. They also introduce a truncated $\beta$-number ${\beta }_{\mu ,2,{\overline{ϵ}}_{\beta }}^k(x,r)$ that is set to zero when $\mu (B_r(x))$ is too small (below ${\overline{ϵ}}_{\beta }{r}^k$). This truncation is crucial for applications where $\mu$ may have arbitrarily small mass on some balls.

Methodologically, the paper develops a multiscale corona decomposition driven by a good/bad dichotomy. At each scale, balls are classified as “good” when the measure has enough mass spread in $k$-dimensional general position (so $\beta$ controls the tilting of best approximating $k$-planes across scales), and “bad” when the measure is effectively concentrated near a $(k-1)$-dimensional affine subspace (so packing estimates compensate for the lack of tilting control). The construction builds sequences of approximating manifolds $T_i$ on good regions using Reifenberg-style interpolation of projections, while bad regions are handled by dimension drop and packing control. The proof is organized around a main technical theorem that produces a decomposition of the ambient region into (i) a packing-controlled set ${\mathcal{C}}^{\prime }$ whose fine-scale part is rectifiable, and (ii) a complementary region where the measure of $\mu$ is quantitatively small.

The principal hypothesis is a Dini-type (square) summability condition on $\beta$-numbers but only in a weak sense: for a covering pair $(\mathcal{C},r_x)$ (which encodes where scale information is available), the set of points where the scale integral of ${\beta }^2$ exceeds $M$ has $\mu$-mass at most $\Gamma$. Concretely, the main theorem assumes $$\mu \{z\in B_1(0):{\int }_{r_z}^2{\beta }_{\mu ,2,{\overline{ϵ}}_{\beta }}^k(z,r{)}^2\frac{dr}{r}>M\}\le \Gamma .$$ Under a smallness requirement on the truncation parameter ${\overline{ϵ}}_{\beta }$ (bounded by a constant depending on $k$, and essentially by $\max \{M,ϵ\}$), the theorem produces a closed subset ${\mathcal{C}}^{\prime }\subset \mathcal{C}$ with four types of conclusions: (A) packing and Minkowski-type bounds for ${\mathcal{C}}^{\prime }$; (B) an upper bound on $\mu$ away from ${\mathcal{C}}^{\prime }$; (C) a noncollapsing lower mass bound on balls centered in ${\mathcal{C}}^{\prime }$; and (D) rectifiable fine-scale structure of ${\mathcal{C}}_0^{\prime }$ with local Hausdorff upper bounds. In particular, the measure control away from ${\mathcal{C}}^{\prime }$ is of the form $$\mu (B_1\setminus B_{r_x}({\mathcal{C}}^{\prime }))\le c(n)(ϵ+M)+\Gamma ,$$ while the packing estimate is scale-uniform: $${\mathcal{H}}^k({\mathcal{C}}_0^{\prime })+\sum _{x\in {\mathcal{C}}_{+}^{\prime }}{r}_x^k\le c(n).$$ These bounds are designed to be effective even without any density assumptions, and the authors emphasize that in general one must split the outcome into a rectifiable, noncollapsed part and a remainder of uniformly bounded $\mu$-mass.

The paper then shows how adding density assumptions converts packing estimates into global mass bounds and yields upper Ahlfors regularity. For example, if one assumes a lower density bound ${\Theta }^{\ast ,k}(\mu ,\mathcal{C},x)\ge a>0$ $\mu$-a.e., then the theorem implies a global Hausdorff bound on the rectifiable part (and even that the complement has $\mu$-measure zero in the covering-pair setting). If instead one assumes an upper density bound ${\Theta }_{\ast }^k(\mu ,\mathcal{C},x)\le b$, then one obtains a global mass bound $$\mu (B_1)\le c(n)({\overline{ϵ}}_{\beta }+b+M)+\Gamma .$$ A further strengthening occurs when $\beta$-estimates hold on all scales (a stronger hypothesis than the one-scale Dini control). Under such all-scales assumptions, the authors derive upper Ahlfors regularity: for every $x$ and $r$ (with appropriate scale restrictions), $\mu (B_r(x))\le c(n)({\overline{ϵ}}_{\beta }+b+M)r^k$. They also obtain rectifiability of $\mu$ when the density is positive $\mu$-a.e. and $\beta$-numbers satisfy one of several integral conditions (A–C in the introduction), including a pointwise-in-$x$ Dini-type bound or a stronger averaged bound.

The paper’s applications include: (i) a decomposition/structure theorem for measures with $L^1$-type $\beta$-control at a scale, yielding a split $\mu ={\mu }_h+{\mu }_{\ell }+{\mu }_0$ where ${\mu }_h$ is supported on a $k$-rectifiable set with volume bounds, ${\mu }_{\ell }$ is $k$-rectifiable with controlled mass, and ${\mu }_0$ has controlled mass and zero upper density; (ii) generalized discrete Reifenberg theorems for discrete measures, where only first-scale $\beta$-control is needed to conclude packing/mass bounds; (iii) a quantitative Reifenberg theorem for sets (when $\mu ={\mathcal{H}}^k└S$); and (iv) an application to boundedness of Calderón–Zygmund operators on measures satisfying a $\beta$-number Carleson-type estimate together with density control.

Limitations are inherent to the generality of the setting. Without density assumptions, the authors show (via examples) that one cannot expect global packing or Minkowski bounds for the entire support: $\beta$-summability alone can hold even for measures with no $k$-dimensional structure (e.g., ${\mathcal{L}}^n$), and one can construct measures where $\beta$-control holds but the support fails any meaningful packing at relevant scales. Consequently, the main results must split the measure into a rectifiable noncollapsed part and a remainder of bounded $\mu$-mass. Even with density assumptions, the paper’s conclusions depend on the availability of $\beta$-control on the relevant scales encoded by the covering pair $(\mathcal{C},r_x)$; in discrete settings, this is essential.

Practically, the results provide a toolkit for turning quantitative flatness information (in the form of $\beta$-numbers) into effective geometric conclusions about measures: bounds on the size of the region where the measure is far from a rectifiable set, and conditions guaranteeing rectifiability and upper Ahlfors regularity. Researchers in geometric measure theory and geometric analysis—especially those studying singular sets in nonlinear PDEs, varifold regularity, and nonhomogeneous harmonic analysis—should care because the framework accommodates holes and excess sets and works for general measures without requiring density or finiteness assumptions beyond those stated.