Result 081, Real and complex analysis

Riesz transforms and rectifiability in higher codimension

Resolves the remaining higher-codimension Riesz-transform rectifiability problem: for d ≥ 4 and $2\le n\le d-2$ , an n-Ahlfors–David regular Radon measure on ℝ^d 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 formalization Proof

The bigger picture

Why it matters

The Riesz transform measures directional interactions across a set. The manuscript claims that controlling this analytic operation forces certain higher-codimension sets to contain substantial, quantitatively controlled pieces parametrized by ordinary Euclidean space, linking an operator bound to geometry.

What changes?

The manuscript reports this for integers d at least 4 and n from 2 through d minus 2. Assume an n-Ahlfors-David regular Radon measure in d-dimensional Euclidean space: balls centered on its support have mass comparable to radius to power n at relevant scales. Uniformly bounded scalar-to-vector L2 (mean-square) norms of its n-dimensional Riesz transform over every positive hard truncation (omitting interactions below a chosen distance) imply uniform n-rectifiability. Quantitative bounds depend only on dimensions, regularity constants and operator bounds.

What does that help mathematicians do?

Uniform rectifiability supplies large pieces of Lipschitz images of n-dimensional balls: parametrizations that stretch distances by at most a fixed factor, with uniform control across locations and scales. The claimed implication would let researchers deduce this geometry from an operator estimate. Conversely, within the stated regularity class and dimensions, failure of that geometry would rule out a uniform bound on the truncated Riesz transforms.

Are there practical applications?

The immediate value is foundational: the result addresses the remaining higher-codimension range of the David-Semmes Riesz-transform question. It would provide a quantitative bridge between singular-integral estimates and geometric structure, allowing an analytic bound to serve as evidence of controlled parametrizations. The supplied abstract describes no practical deployment or computational algorithm.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Riesz transforms and uniform rectifiability in higher codimension

September 24, 2026 56 pages Main result formalized in Lean

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.

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Cite (BibTeX)

@misc{OAI:Riesz-transforms-and-uniform-rectifiability-in-higher-codimension-September-24-2026,
  author = {{OpenAI}},
  title = {{Riesz transforms and uniform rectifiability in higher codimension}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Riesz-transforms-and-uniform-rectifiability-in-higher-codimension-September-24-2026/paper.pdf}{OAI:Riesz-transforms-and-uniform-rectifiability-in-higher-codimension-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/081.md.

Riesz transforms and rectifiability in higher codimension

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Riesz transforms and uniform rectifiability in higher codimension

Scope

The formalized result proves that bounded Riesz transforms force quantitative uniform rectifiability in higher codimension. For $d\ge4$ and $2\le n\le d-2$ , an $n$ -Ahlfors–David regular measure whose positive hard truncations have one uniform $L^2$ operator bound has big pieces of Lipschitz images. The mass fraction and Lipschitz constant depend only on the dimensions and the stated regularity and operator bounds, and work at every support point and admissible radius, including unbounded support. The variation and principal-value corollary is not included.

Comparator links

Result	Comparator statement
Quantitative higher-codimension Riesz rectifiability	RieszQuantitative.lean

At a glance

Subject: Real and complex analysis
Manuscripts: 1
Pages: 56
Dated: Sep 24, 2026
Lean formalization: Yes
Reasoning summary: No

Named problems addressed

David–Semmes Riesz-transform question in its remaining higher-codimension rangeProved
David–Semmes Riesz-transform questionPartial progress

Indexed by Emergent Mind with an LLM from the text above; check the manuscripts for the precise statements.

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