The weak lower density condition and uniform rectifiability

Abstract

We show that an Ahlfors \(d\)-regular set \(E\) in \(\mathbb{R}^{n}\) is uniformly rectifiable if the set of pairs \((x,r)\in E\times (0,\infty)\) for which there exists \(y \in B(x,r)\) and \(0<t<r\) satisfying \(\mathbf{H}^{d}_{\infty}(E\cap B(y,t))<(2t)^{d}-\epsilon(2r)^d\) is a Carleson set for every \(\epsilon>0\). To prove this, we generalize a result of Schul by proving, if \(X\) is a \(C\)-doubling metric space, \(\epsilon,\rho\in (0,1)\), \(A>1\), and \(X_n\) is a sequence of maximal \(2^{-n}\)-separated sets in \(X\), and \(\mathbf{B}=\{B(x,2^{-n})\colon x\in X_{n},n\in \mathbb{N}\}\), then   \(\sum \left\{r_{B}^s\colon B\in \mathbf{B}, \frac{\mathbf{H}^s_{\rho r_{B}}(X\cap AB)}{(2r_{AB})^s}>1+\epsilon\right\} \le_{C,A,\epsilon,\rho,s} \mathbf{H}^s(X)\).
This is a quantitative version of the classical result that for a metric space \(X\) of finite \(s\)-dimensional Hausdorff measure, the upper \(s\)-dimensional densities are at most 1 \(\mathbf{H}^s\)-almost everywhere.