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 $(x,r)\in E\times (0,\mathrm{\infty })$ for which there exists $y\in B(x,r)$ and $0<t<r$ satisfying ${\mathbf{H}}_{\mathrm{\infty }}^d(E\cap B(y,t))<(2t{)}^d-ϵ(2r{)}^d$ is a Carleson set for every $ϵ>0$. To prove this, we generalize a result of Schul by proving, if $X$ is a $C$-doubling metric space, $ϵ,\rho \in (0,1)$, $A>1$, and $X_n$ is a sequence of maximal $2^{-n}$-separated sets in $X$, and $\mathbf{B}=\{B(x,2^{-n}):x\in X_n,n\in \mathbb{N}\}$, then   $\sum \{r_B^s:B\in \mathbf{B},\frac{{\mathbf{H}}_{\rho r_B}^s(X\cap AB)}{(2r_{AB}{)}^s}>1+ϵ\}{\le }_{C,A,ϵ,\rho ,s}{\mathbf{H}}^s(X)$.
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.