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 ${\mathcal{H}}_{\infty }^d(E\cap B(y,t))<(2t{)}^d-\epsilon (2r{)}^d$ 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 $\mathcal{B}=\{B(x,2^{-n}):x\in X_n,n\in \mathbb{N}\}$, then \[ \sum \left\{r_{B}^{s}: B\in \mathscr{B}, \frac{\mathscr{H}^{s}_{ρr_{B}}(X\cap AB)}{(2r_{B})^{s}}>1+\varepsilon\right\} \lesssim_{C,A,\varepsilon,ρ,s} \mathscr{H}^{s}(X). \] 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 $1$ ${\mathcal{H}}^s$-almost everywhere.

Fixed typos and minor errors, expanded the introduction, and corrected the counterexample on page 5