The endpoint case of Carleson's pointwise convergence problem for the Schrodinger equation: a.e. convergence for data in H^{n/(2(n+1))}

Carleson asked for which Sobolev exponents $s$ the free Schrodinger evolution $e^{it\Delta}f$ converges to $f$ almost everywhere as $t\to0$ for every $f\in H^s(\mathbb R^n)$ . In one dimension the answer is $s\ge1/4$ (Carleson; Dahlberg-Kenig). Bourgain's 2016 counterexample showed that $s\ge n/(2(n+1))$ is necessary for $n\ge2$ but did not rule out equality; Du, Guth and Li (planar) and Du and Zhang ( $n\ge3$ ) proved convergence for every $s>n/(2(n+1))$ , leaving only the endpoint. Does $e^{it\Delta}f\to f$ almost everywhere for every $f\in H^{n/(2(n+1))}(\mathbb R^n)$ , $n\ge2$ (in the plane, for $H^{1/3}$ )?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Harmonic analysis; dispersive equations and pointwise convergence
Posed by: Lennart Carleson (the convergence problem); the endpoint left open after Bourgain (2016), Du-Guth-Li (2017) and Du-Zhang (2019)
Year posed: 1980
Years open: 46y
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 32 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Planar theorem: for every $f\in H^{1/3}(\mathbb R^2)$ there is a full-measure set $E_f$ on which the Gaussian-regularised evolution has a limit for every $0<t<1$ and converges to $f$ as $t\downarrow0$ . Higher-dimensional theorem: the same for $f\in H^{n/(2(n+1))}(\mathbb R^n)$ , every $n\ge3$ . With Bourgain's necessary condition this gives the exact range $s\ge n/(2(n+1))$ for $n\ge2$ . Not shown: the endpoint local $L^3$ maximal inequality in the plane, sharp $L^p$ maximal estimates at the endpoint in higher dimensions, or results for other dispersive equations or convergence along curves.

What the AI did

The release README says the vast majority of results were obtained with one fixed procedure using an unreleased internal OpenAI model, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier results produced by the models. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human edited). The manuscripts are authored 'OpenAI' and name no human author. The family has two manuscripts dated September 24, 2026: the planar endpoint $H^{1/3}(\mathbb R^2)$ and the endpoint in every dimension $n\ge3$ .

Verification

No independent mathematician has checked this yet. Checked here: the main theorems of both manuscripts were read against Carleson's problem and the endpoint left by Bourgain and Du-Zhang. The evolution is defined by taking the Gaussian-regularisation limit first, on one full-measure spatial set and for all $0<t<1$ , which the papers state explicitly; this is a standard way to give meaning to $e^{it\Delta}f$ pointwise. The planar paper says its maximal estimate has local $L^1$ output and that the stronger endpoint local $L^3$ maximal estimate is not asserted. Not refereed; no Lean formalization.

The endpoint case of Carleson's pointwise convergence problem for the Schrodinger equation: a.e. convergence for data in H^{n/(2(n+1))}

What was actually shown

What the AI did

Verification

Sources

Discussion