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In this paper we show that if (mu ) is a Borel measure in ({{mathbb {R}}}^{n+1}) with growth of order n, such that the n-dimensional Riesz transform ({{mathcal {R}}}_mu ) is bounded in (L^2(mu )), and (Bsubset {{mathbb {R}}}^{n+1}) is a ball with (mu (B)approx r(B)^n) such that:(a)there is some n-plane L passing through the center of B such that for some (delta u003e0) small enough, it holds $$begin{aligned}int _B frac{mathrm{dist}(x,L)}{r(B)},dmu (x)le delta ,mu (B),end{aligned}$$(b)for some constant ({varepsilon }u003e0) small enough, $$begin{aligned}int _{B} |{{mathcal {R}}}_mu 1(x) - m_{mu ,B}({{mathcal {R}}}_mu 1)|^2,dmu (x) le {varepsilon },mu (B),end{aligned}$$where (m_{mu ,B}({{mathcal {R}}}_mu 1)) stands for the mean of ({{mathcal {R}}}_mu 1) on B with respect to (mu ),then there exists a uniformly n-rectifiable set  ), with (mu (Gamma cap B)gtrsim mu (B)), and such that (mu |_Gamma ) is absolutely continuous with respect to ({{mathcal {H}}}^n|_Gamma ). 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.

The Riesz transform and quantitative rectifiability for general Radon measures