Result 075, Real and complex analysis

The Llog⁡LL\log L Fourier-convergence conjecture

Proves that the ordinary symmetric Fourier partial sums of every complex-valued function in Llog⁡L(T)L\log L(\mathbb T) converge almost everywhere along the full sequence. This resolves the classical sufficiency conjecture at the Llog⁡LL\log L scale.

Proof

The bigger picture

Why it matters

Fourier series reconstruct a periodic function by adding its frequency components. This unreviewed manuscript claims that a specific integrability condition guarantees convergence at almost every point, even when the function is not smooth.

What changes?

The claimed result applies to every complex-valued function on the one-dimensional circle in L log L: the integral of its magnitude times the logarithm of two plus its magnitude is finite. Its ordinary symmetric Fourier partial sums, which retain all integer frequencies from minus N through N, converge to the function outside a set of measure zero as N increases through all positive integers. No averaging or selected subsequence is required. The manuscript reports resolving the classical L log L sufficiency conjecture.

What does that help mathematicians do?

The result would let researchers deduce almost-everywhere Fourier reconstruction directly from this integrability test, without imposing smoothness. It would also rule out a function in this class whose ordinary symmetric partial sums diverge on a set of positive measure. That identifies a sufficient condition for convergence, not a necessary one: the claim neither settles every function outside L log L nor guarantees convergence at every point.

Are there practical applications?

Its immediate value is foundational: it clarifies when frequency expansions recover a periodic function point by point, apart from a negligible exceptional set. Fourier methods also underlie practical signal representations, but this claim alone supplies no convergence rate or finite-sum error bound. It therefore does not establish how many frequencies a numerical reconstruction would need.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Almost-everywhere Fourier convergence in L log L

September 23, 2026 76 pages

We prove that the ordinary symmetric Fourier partial sums of every complex-valued function in Llog⁡LL\log L on the circle converge almost everywhere to the function along the full sequence. This establishes the classical Llog⁡LL\log L sufficiency conjecture.

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Cite (BibTeX) 
@misc{OAI:Almost-everywhere-Fourier-convergence-in-L-log-L-September-23-2026,
  author = {{OpenAI}},
  title = {{Almost-everywhere Fourier convergence in $L\log L$}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Almost-everywhere-Fourier-convergence-in-L-log-L-September-23-2026/paper.pdf}{OAI:Almost-everywhere-Fourier-convergence-in-L-log-L-September-23-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.