The weak lower density condition and uniform rectifiability

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 \(01+\epsilon\right\} \le_{C,A,\epsilon,\rho,s} \mathbf{H}^s(X)\). 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.