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The David-Semmes problem in higher codimension: bounded Riesz transforms force uniform rectifiability for 1 < n < d-1

Theorem 1.3: for integers $d\ge4$ and $2\le n\le d-2$ , an $n$ -AD-regular Radon measure on $\mathbb R^d$ whose hard-truncated Riesz transforms are uniformly bounded $L^2(\mu)\to L^2(\mu;\mathbb R^d)$ is uniformly $n$ -rectifiable, with constants depending only on $d,n,C_{AD},C_R$ ; the conclusion is given as uniform big pieces of Lipschitz images of balls. With the known cases $n=1$ and $n=d-1$ this answers the David-Semmes question in every dimension and codimension. The converse direction and non-integer dimensions are not addressed here.

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Occurred: Sep 24, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The David-Semmes problem in higher codimension: bounded Riesz transforms force uniform rectifiability for 1 < n < d-1 · David-Semmes problem, higher codimension

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Unreleased internal OpenAI model
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VibeMathed record: The David-Semmes problem in higher codimension: bounded Riesz transforms force uniform rectifiability for 1 < n < d-1

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No independent mathematician has checked this yet. Checked here: Theorem 1.3 was read against the David-Semmes question in the range left open; for $d\ge4$ and $2\le n\le d-2$ (all of $1<n<d-1$ ) it gives uniform rectifiability from a uniform $L^2$ bound on hard truncations, with no principal-value or variation hypothesis. The proof was not refereed. lean/formalization.yaml lists RieszQuantitative (OAI.RieszRectifiability.quantitative_higher_codimension_riesz_rectifiability). The comparator statement was read here: for such $d,n$ and constants $C_{AD}$ , $C_R$ there are $\theta>0$ and $M$ such that every regular $n$ -AD-regular measure with truncations bounded by $C_R$ has, at every support point and admissible radius, an $M$ -Lipschitz image of the $n$ -ball of radius $r$ carrying mass at least $\theta r^n$ in the ball. This states the headline (big pieces of Lipschitz images, the manuscript's definition of uniform rectifiability). Not rebuilt here.

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The David-Semmes problem in higher codimension: bounded Riesz transforms force uniform rectifiability for 1 < n < d-1

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Lean proof (OAI.RieszRectifiability.quantitative_higher_codimension_riesz_rectifiability)

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Comparator statement: RieszQuantitative.lean

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Lean proof (OAI.RieszRectifiability.quantitative_higher_codimension_riesz_rectifiability)

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Comparator statement: RieszQuantitative.lean

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This event attributed to Unreleased internal OpenAI model

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Comparator statement: RieszQuantitative.lean evidence for this event

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