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Math-2397-A1 · revision r1 · on Math-2397

OpenAI claim summary for Math-2397

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

Fourier convergence at the Orlicz endpoint: For a complex function on the circle satisfying the $L\log L$ integrability condition, the paper claims convergence of its symmetric Fourier sums almost everywhere. The limit is asserted for the ordinary full sequence of partial sums. The parameters, regularity assumptions and normalization in this question are retained in applying that assertion.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. Almost-Everywhere Fourier Convergence in \$L^{1}\$
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

Almost-Everywhere Fourier Convergence in \$L^{1}\$

OpenAI's claim

Fourier convergence at the Orlicz endpoint: For a complex function on the circle satisfying the $L\log L$ integrability condition, the paper claims convergence of its symmetric Fourier sums almost everywhere. The limit is asserted for the ordinary full sequence of partial sums.

Mathematical statement of the claimed result

Mathematical claim 1.
On $\mathbb T=\mathbb R/(2\pi\mathbb Z)$ with normalized measure $\mu$, suppose $\int|f|\log(2+|f|)\,d\mu<\infty$. Set $\widehat f(k)=\int f(x)e^{-ikx}\,d\mu(x)$ and $S_Nf=\sum_{k=-N}^N\widehat f(k)e^{ikx}$. The claimed conclusion is \[S_Nf(x)\longrightarrow f(x)\quad\text{for almost every }x,\quad N=0,1,2,\ldots.\] It applies to complex-valued inputs and the full sequence of symmetric partial sums, under the stronger $L\log L$ hypothesis rather than merely $L^1$.

Outline of the argument

Fourier convergence at the Orlicz endpoint: A bounded nonnegative density is first controlled on the set where its spatial maximal average is moderate. The corresponding modulated dyadic sums are stopped on a finite binary tree, giving a good-set integral estimate uniform in the depth. Averaging delayed Haar kernels recovers the Hilbert kernel; doubly exponential height layers then make both the integral costs and exceptional-set measures summable.

Formal verification

Fourier convergence at the Orlicz endpoint: The pinned catalogue lists no Lean formalization for this result.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources