Riesz transforms and rectifiability in higher codimension

What OpenAI claims

Resolves the remaining higher-codimension Riesz-transform rectifiability problem: for d ≥ 4 and 2 ≤ n ≤ d-2, an n-Ahlfors–David regular Radon measure on ℝd is uniformly n-rectifiable whenever its n-dimensional Riesz transform is uniformly L2-bounded over all positive hard truncations. The rectifiability bounds depend only on dimension, regularity and operator bounds. (Lean)

Paper

1. Riesz transforms and uniform rectifiability in higher codimension

   24 September 2026 · Preprint, not peer reviewed · PDF

   OpenAI's abstract We prove that an n-Ahlfors–David regular Radon measure on ℝd is uniformly n-rectifiable if its n-dimensional Riesz transform is uniformly bounded from scalar L²(μ) to vector-valued L²(μ) over all positive hard truncations, for integers d ≥ 4 and 2 ≤ n ≤ 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.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.