Hannah Cairo (2025), arXiv preprint — ARXIV-2502.06137
Key takeaways
- For every C² hypersurface in ℝ^d that is not contained in a hyperplane, there are f and a nonnegative weight w such that ∫_{B_R}|Ef|²w ≳ log R·‖f‖²·sup_ℓ∫_ℓ w. So the Mizohata–Takeuchi conjecture fails (with log R loss), Stein’s conjecture as stated fails, and the Mizohata–Takeuchi route to loss-free endpoint multilinear restriction is closed.
Seeded from the abstract and a skim, before the reading in NOTE-051. What follows is what the work says about itself.
The paper proves a family of L^p estimates for the X-ray transform of positive measures in ℝ^d. It uses them to build a counterexample to the Mizohata–Takeuchi conjecture, with a log R loss, for every C² hypersurface that does not lie in a hyperplane. One consequence is that the endpoint multilinear restriction estimates cannot be sharpened directly via Mizohata–Takeuchi.
Standing in the record
Filed from the survey of 2026-09-25 of work the anthology set aside as out of scope (tier C): 335 seconds of active reading over 2 sessions in the papers-feed tracker. It was filed Deferred, unread. NOTE-051 is the close reading of 2026-09-25, and it placed the work: Active — a short, readable disproof of a 40-year-old conjecture in Fourier restriction. The counting argument is checkable by a non-specialist; the incidence lemma (§4) is the part that needs an expert. It is worth reading as mathematics. The “local” replacement it proposes (Conj. 1.5) has since been refuted for a dense family of C^k hypersurfaces by Cairo–Zhang (arXiv:2512.08064), so it should be read together with that paper.
When seeded, before the reading in NOTE-051, the entry gave a priority for a deeper reading: low — The owner has read it for moderate time (t=335, n=2). The skim captures the result and its significance, and the remaining content is technical and off-scope for the anthology.
What the seeding asked a deeper reading to check (NOTE-051 is that reading):
- It is pure harmonic analysis with no direct ML connection. It matters for the reading list only as a notable result: a decades-old conjecture refuted by a short argument.
- Check the status of the local conjecture (Conj. 1.5) and whether a journal version or follow-ups have strengthened the loss beyond log R.
Access when seeded: arXiv abs page (v1 10 Feb 2025, v2 12 Mar 2025) and full PDF of v2 (15 pp.) read directly: abstract, all of §1 (§1.1–1.8), and the opening of the §3 construction. A Crossref bibliographic search found no journal version, so the venue is arXiv as of this check.