In this paper we show that if $$\mu $$ μ is a Borel measure in $${{\mathbb {R}}}^{n+1}$$ Rn+1 with growth of order n, such that the n-dimensional Riesz transform $${{\mathcal {R}}}_\mu $$ Rμ is bounded in $$L^2(\mu )$$ L2(μ) , and $$B\subset {{\mathbb {R}}}^{n+1}$$ B⊂Rn+1 is a ball with $$\mu (B)\approx r(B)^n$$ μ(B)≈r(B)n such that:(a)
there is some n-plane L passing through the center of B such that for some $$\delta >0$$ δ>0 small enough, it holds $$\begin{aligned}\int _B \frac{\mathrm{dist}(x,L)}{r(B)}\,d\mu (x)\le \delta \,\mu (B),\end{aligned}$$ ∫Bdist(x,L)r(B)dμ(x)≤δμ(B),
(b)
for some constant $${\varepsilon }>0$$ ε>0 small enough, $$\begin{aligned}\int _{B} |{{\mathcal {R}}}_\mu 1(x) - m_{\mu ,B}({{\mathcal {R}}}_\mu 1)|^2\,d\mu (x) \le {\varepsilon }\,\mu (B),\end{aligned}$$ ∫B|Rμ1(x)-mμ,B(Rμ1)|2dμ(x)≤εμ(B), where $$m_{\mu ,B}({{\mathcal {R}}}_\mu 1)$$ mμ,B(Rμ1) stands for the mean of $${{\mathcal {R}}}_\mu 1$$ Rμ1 on B with respect to $$\mu $$ μ ,
then there exists a uniformly
n-rectifiable set $$\Gamma $$ Γ , with $$\mu (\Gamma \cap B)\gtrsim \mu (B)$$ μ(Γ∩B)≳μ(B) , and such that $$\mu |_\Gamma $$ μ|Γ is absolutely continuous with respect to $${{\mathcal {H}}}^n|_\Gamma $$ Hn|Γ . This result is an essential tool to solve an old question on a two phase problem for harmonic measure in subsequent papers by Azzam, Mourgoglou, Tolsa, and Volberg.