Uniform rectifiability from Carleson measure estimates and $\epsilon$-approximability of bounded harmonic functions

Let $\mathrm{\Omega }\subset {\mathbb{R}}^{n+1}$, $n\ge 1$, be a corkscrew domain with Ahlfors-David regular boundary. In this paper we prove that $\mathrm{\partial }\mathrm{\Omega }$ is uniformly $n$-rectifiable if every bounded harmonic function on $\mathrm{\Omega }$ is $\epsilon$-approximable or if every bounded harmonic function on $\mathrm{\Omega }$ satisfies a suitable square-function Carleson measure estimate. In particular, this applies to the case when $\mathrm{\Omega }={\mathbb{R}}^{n+1}\setminus E$ and $E$ is Ahlfors-David regular. Our results solve a conjecture posed by Hofmann, Martell, and Mayboroda in a recent work where they proved the converse statements. Here we also obtain two additional criteria for uniform rectifiability. One is given in terms of the so-called “$S<N$” estimates, and another in terms of a suitable corona decomposition involving harmonic measure.