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Almost everywhere convergence of Fourier series in L log L

For an integrable function ff on the circle T\mathbb T let SNf(x)=∑∣k∣≤Nf^(k)eikxS_Nf(x)=\sum_{|k|\le N}\widehat f(k)e^{ikx}. Carleson (1966) proved SNf→fS_Nf\to f almost everywhere for f∈L2f\in L^2, Hunt (1968) extended this to LpL^p, p>1p>1, and Kolmogorov (1923) gave an integrable function whose Fourier series diverges almost everywhere. Between these, convergence was known in Llog⁡Llog⁡log⁡LL\log L\log\log L (Sjolin 1969), in Llog⁡Llog⁡log⁡log⁡LL\log L\log\log\log L (Antonov 1996) and in the quasi-Banach space QA\mathrm{QA} of Arias de Reyna. Does SNf(x)→f(x)S_Nf(x)\to f(x) for almost every xx, along the full sequence N=0,1,2,…N=0,1,2,\ldots, for every ff in the Orlicz class Llog⁡L(T)={f:∫∣f∣log⁡(2+∣f∣)<∞}L\log L(\mathbb T)=\{f:\int|f|\log(2+|f|)<\infty\}?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Harmonic analysis; pointwise convergence of Fourier series
Posed by
Classical question after Carleson, Hunt, Sjolin and Antonov; stated as Conjecture 1 (with a weak-L1 maximal formulation) by Victor Lie (2017)
Year posed
—
Years open
—
Solved
2026-09-23
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
52 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for every complex-valued f∈Llog⁡L(T)f\in L\log L(\mathbb T) there is a null set NfN_f with lim⁡N→∞SNf(x)=f(x)\lim_{N\to\infty}S_Nf(x)=f(x) for x∉Nfx\notin N_f, along the full sequence. The key estimate is a good-set bound ∫{Mρ≤e4h}sup⁡N∣SNρ∣≤Cd\int_{\{M\rho\le e^{4h}\}}\sup_N|S_N\rho|\le Cd for bounded densities of height ede^d, summed over height layers. The paper also deduces a weak-L1L^1 maximal inequality in Luxemburg norm. It does not address spaces strictly between Llog⁡LL\log L and L1L^1 (where Konyagin's divergence results apply below Llog⁡LL\sqrt{\log L}), nor lacunary or higher-dimensional analogues.

What the AI did

The release README says every result in it was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. 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). The manuscript is authored 'OpenAI' and names no human author. The manuscript (September 23, 2026) has no companion in the release.

Verification

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∈Llog⁡L(T)f\in L\log L(\mathbb T), which is the question as posed. A corollary also gives the Luxemburg-norm weak-L1L^1 maximal bound via Stein's theorem. The proof (an entropy-compression argument for a dyadic model, transferred to Fourier sums by dyadic representation of the Hilbert kernel) was not refereed. No Lean formalization exists for this manuscript in the release.

Sources

Changelog1 change

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