Enough to converge?

0:00Do Fourier series converge?

A periodic function can be written as a sum of sines and cosines. Cut the sum off at frequency N, and you get the partial sum S_N f. Do the partial sums converge back to f? For a square wave they do, at every point away from the jumps, though near a jump they overshoot by about 9% of the jump. But in 1923, Kolmogorov built an integrable function whose Fourier series diverges almost everywhere, and a few years later one that diverges everywhere. In 1966, Carleson proved that the Fourier series of every square-integrable function converges almost everywhere, and Hunt extended this to Lᵖ for every p greater than one. Between Lᵖ and L¹ lies a ladder of classes, measured by how much more than the absolute value of f is integrable. Sjölin reached L log L log log L in 1969, and Antonov L log L log log log L in 1996; Soria found other convergence spaces by extrapolation, and Arias de Reyna and others later found classes larger than Antonov’s. In the other direction, Konyagin built everywhere divergent examples in every class a little smaller than L√(log L / log log L). The conjecture in between: L log L is enough. That is the class of functions for which the absolute value of f, times log of the absolute value of f, is integrable.

1:30The claim

Result 075 is one OpenAI preprint, 76 pages, dated September 23. It claims to prove the conjecture: for every complex-valued function in L log L on the circle, the ordinary symmetric partial sums converge to f almost everywhere, along the full sequence of N. As a consequence, the largest partial sum obeys a weak-type bound in terms of the L log L norm. OpenAI also published a Lean proof of the theorem: over 150 files and about 30,000 lines, matched against a formal statement in the release. We have not run it.

2:09Why L log L

The proof rests on one estimate. Take a nonnegative input ρ with total mass one and values below e^d, for large d. Remove the points where its maximal average exceeds (log d)⁴; the removed set has measure at most a constant over (log d)⁴. On the rest, the integral of the largest partial sum is at most a constant times d. Why does that give L log L? Split f into layers by height: values between e^(2^j) and e^(2^(j+1)). A layer of mass m, rescaled to mass one, has height about e^d and costs about m times d, and adding mass times the log of the height over all layers is essentially the integral of the absolute value of f times its log. For a function in L log L, the costs shrink and their sum is finite. For a function like this one, just outside L log L, every layer costs about the same, and the sum diverges. The removed sets shrink at least as fast as 1/j⁴, and those measures have a finite sum. So almost every point is removed only finitely often. Off the removed sets, the integral bound and the finite total cost mean that, at almost every point, the largest partial sums of the far layers add up to something finite. The few remaining layers are bounded functions, which the same estimate handles. That is where the conclusion comes from.

3:36Trees, noise and phases

To prove the estimate, the paper works on a binary tree of intervals. Follow a path down the tree. At each interval, take half the difference of the averages of the modulated input on the two halves, and emit it with the sign of a later left-or-right step. Stop the path when the input’s average first exceeds the threshold. The key proposition bounds the integral of the largest such sum, over all integer frequencies, by a constant times d, however deep the tree and however many frequencies. Handling many frequencies at once, without a loss that grows with their number, is where the new ideas enter. The paper observes the input point through noise: a label equal to the point plus a random shift, whose density is proportional to e^(−A), with A a nonnegative combination of one-minus-cosine terms. An entropy compression lemma finds a large set of frequencies whose phases nearly agree on one region. Correlation inequalities of Griffiths and Ginibre help control what each new observation costs. Counted on plateaus of the tree, the costs come to order d/h on each, with total weight order h, so order d overall, where h is log log d. Finally, averaging these tree kernels over shifts and scales of the tree gives a nonzero multiple of the Hilbert kernel, 1/(x − y), up to an error controlled on the good set, following Petermichl and Hytönen. The Hilbert transform then produces the ordinary Fourier partial sums.

5:10Where it stands

The preprint is not yet peer reviewed, and it has not been revised. OpenAI published a Lean proof; we have not run the check. The paper cites no other release paper. Its outside inputs are published results: Chang’s method for large spectra in Lee’s entropy form, the Griffiths and Ginibre inequalities, Ville’s maximal inequality, the dyadic averaging of Petermichl and of Hytönen, and Stein’s theorem for the maximal bound. We have seen no expert review. If it holds, the classical question at the L log L scale is settled: every function in L log L has an almost everywhere convergent Fourier series, along the full sequence of partial sums. What happens in the classes between Konyagin’s divergent examples and L log L remains open.