  Suppose that $\Omega \subset \mathbb{R}^{n+1}$, $n \ge 2$, is an open set
satisfying the corkscrew condition with an $n$-dimensional ADR boundary,
$\partial \Omega$. In this note, we show that if harmonic functions are
$\varepsilon$-approximable in $L^p$ for any $p > n/(n-1)$, then $\partial
\Omega$ is uniformly rectifiable. Combining our results with those in [HT]
(Hofmann-Tapiola) gives us a new characterization of uniform rectifiability
which complements the recent results in [HMM] (Hofmann-Martell-Mayboroda),
[GMT] (Garnett-Mourgoglou-Tolsa) and [AGMT] (Azzam-Garnett-Mourgoglou-Tolsa).
