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Almost everywhere convergence of Fourier series in L log L

Theorem 1.1: for every complex-valued $f\in L\log L(\mathbb T)$ there is a null set $N_f$ with $\lim_{N\to\infty}S_Nf(x)=f(x)$ for $x\notin N_f$ , along the full sequence. The key estimate is a good-set bound $\int_{\{M\rho\le e^{4h}\}}\sup_N|S_N\rho|\le Cd$ for bounded densities of height $e^d$ , summed over height layers. The paper also deduces a weak- $L^1$ maximal inequality in Luxemburg norm. It does not address spaces strictly between $L\log L$ and $L^1$ (where Konyagin's divergence results apply below $L\sqrt{\log L}$ ), nor lacunary or higher-dimensional analogues.

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Prior state unknown→proved

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Occurred: Sep 23, 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: Almost everywhere convergence of Fourier series in L log L · Fourier convergence in L log L

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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VibeMathed
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Unreleased internal OpenAI model
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VibeMathed record: Almost everywhere convergence of Fourier series in L log L

specialist registry · passed

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the manuscript (TeX source) was read against the question; it states almost-everywhere convergence of the ordinary symmetric partial sums along the full sequence for every complex-valued $f\in L\log L(\mathbb T)$ , which is the question as posed. Lean-checked on the release's Comparator challenge FourierLLogL (declaration OAI.FourierLLogL.paperMain, solution module OAI.Analysis.FourierLLogL.PaperMain), added in the 7 October 2026 update and listed in lean/formalization.yaml. Its statement was read here: for every a.e.-strongly measurable $f$ on $\mathbb R/2\pi\mathbb Z$ with $\int|f|\log(2+|f|)<\infty$ (Haar measure) there is a measurable null set outside which $\sum_{|k|\le N}\widehat f(k)e^{ikx}\to f(x)$ as $N\to\infty$ . That is the headline. Not rebuilt here. Permitted axioms: propext, Quot.sound, Classical.choice. The corollary's Luxemburg-norm weak- $L^1$ maximal bound is not part of the Comparator statement. The proof (an entropy-compression argument for a dyadic model, transferred to Fourier sums) was not refereed.

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Almost everywhere convergence of Fourier series in L log L

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Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Lie (2017), Pointwise convergence of Fourier series (I), Conjecture 1

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problem-record link supplied by the VibeMathed registry.

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Lean Comparator challenge FourierLLogL (OpenAI math release)

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lean-proof link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean Comparator challenge FourierLLogL (OpenAI math release)

lean artifact · pending

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VibeMathed record: Almost everywhere convergence of Fourier series in L log L evidence for this event

Lean Comparator challenge FourierLLogL (OpenAI math release) evidence for this event

This event attributed to Unreleased internal OpenAI model

Almost everywhere convergence of Fourier series in L log L evidence for this event

Almost everywhere convergence of Fourier series in L log L parent of this event

Lie (2017), Pointwise convergence of Fourier series (I), Conjecture 1 evidence for this event

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