Browse › Real and complex analysis › 081 Riesz transforms and rectifiability in higher codimension
Riesz transforms and uniform rectifiability in higher codimension
Open PDFCite Report a problem Add a reference or comment Interested in working on this Start a discussion
Edit NOTES.md · Existing issues · Source on GitHub
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 $L^2(\mu)$ 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.
If the PDF does not display here, open it directly: Open PDF.
Notes
No notes yet. Add a reference, a comment or your name with the buttons above, or edit NOTES.md on GitHub.
Automatically collected notes (unreviewed)
work in progress
Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- John Hoffman and Benjamin Jaye, On Singular Integrals and Quantitative Rectifiability in Parabolic Space and the Heisenberg Group (arXiv:2510.26934v4, 10 Oct 2026). The v4 introduction calls the intermediate cases "the longstanding higher-codimension David–Semmes problem", which "a recent preprint ... claims" to resolve positively, and says a version of the authors' technique "also appears in OpenAI's recent claimed solution".
Formalization
- OpenAI math release, Lean scope notes for family 081 (lean/docs/081.md, 2026-10-06). Gives the scope of the release's Lean formalization: the quantitative big-pieces-of-Lipschitz-images statement for $d\ge4$, $2\le n\le d-2$, under AD-regularity and uniform hard-truncation bounds (including unbounded support), without the variation and principal-value corollary. [family-level]
- OpenAI math release, RieszQuantitative.lean (comparator statement, Oct 2026). The comparator statement
QuantitativeFullStatement: AD-regularity and a uniform $L^2$ bound for the truncations imply the ball-image conclusion, with constants $\theta, M$ chosen before the measure. - OpenAI math release, Rectifiability.lean (Lean source, Oct 2026). Contains the theorem
quantitative_higher_codimension_riesz_rectifiability, which proves the comparator statement.
Possibly relevant literature (found by the agent, not checked)
- Daniel Girela-Sarrión and Xavier Tolsa, The Riesz transform and quantitative rectifiability for general Radon measures (arXiv:1601.08079, 2016–2017). A codimension-one result in which closeness to a plane plus small $L^2$ oscillation of the Riesz transform on a ball gives a big uniformly rectifiable piece; it may be relevant to the paper's flatness-propagation and small-oscillation mechanism (Sections 6–7).
- Xavier Tolsa, New criteria for the rectifiability of Radon measures in terms of Riesz transforms (arXiv:2512.14534, Dec 2025, rev. Feb 2026). Codimension-one criteria in which small oscillation of $R\mu$ on a doubling ball gives a big uniformly rectifiable piece; the abstract states this for $\mathbb R^{n+1}$ only, which is relevant to how far such oscillation criteria were known before the paper's higher-codimension argument.
- Benjamin Jaye and Fedor Nazarov, Reflectionless measures for Calderón–Zygmund operators II: Wolff potentials and rectifiability (arXiv:1507.08329, 2015-07-29). Its reduction of geometric consequences of operator boundedness to rigidity of reflectionless measures is potentially relevant to the manuscript's compactness-and-rigidity route from bounded Riesz transforms to uniform rectifiability.
Digests and expositions
- Gradient Damage, Falconer Distance Conjecture, Bochner–Riesz Conjecture and 8 More (video compilation, 2026-10-15). The chapter beginning at 2:48 introduces Riesz transforms and rectifiability through a force analogy and presents the release's higher-codimension claim as an illustration rather than a proof. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 081
Independent sites; their content is not reviewed here.
Cite
@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 provides Lean files for this family: lean/docs/081.md. The Lean statements have not been checked by people against the paper's claims.
Family 081: 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 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.
This is the only paper in this family.