Uniform rectifiability and $\epsilon$-approximability of harmonic functions in $L^p$

Suppose that $E\subset {\mathbb{R}}^{n+1}$ is a uniformly rectifiable set of codimension $1$. We show that every harmonic function is $\epsilon$-approximable in $L^p(\mathrm{\Omega })$ for every $p\in (1,\mathrm{\infty })$, where $\mathrm{\Omega }:={\mathbb{R}}^{n+1}\setminus E$. Together with results of many authors this shows that pointwise, $L^{\mathrm{\infty }}$ and $L^p$ type $\epsilon$-approximability properties of harmonic functions are all equivalent and they characterize uniform rectifiability for codimension $1$ Ahlfors-David regular sets. Our results and techniques are generalizations of recent works of T. Hyt"onen and A. Ros'en and the first author, J. M. Martell and S. Mayboroda.