Riesz transforms and uniform rectifiability in higher codimension

We prove that an n-Ahlfors–David regular Radon measure on ℝ${}^d$ is uniformly n-rectifiable if its n-dimensional Riesz transform is uniformly bounded from scalar $L^2(\mu )$ to vector-valued $L^2(\mu )$ over all positive hard truncations, for integers d ≥ 4 and $2\le n\le d-2$. This gives a positive answer to the David–Semmes Riesz-transform question in its remaining higher-codimension range. The conclusion gives uniform big pieces of Lipschitz images of Euclidean balls.