Flat with signs only?

0:00Signs on a circle

Take N coefficients, each plus one or minus one, and form the polynomial with those coefficients. This is a Littlewood polynomial. On the unit circle, the average of its squared size is exactly N, so its maximum is at least √N. How close to that can the maximum be? A random choice of signs at length 1,024 peaks near three times √N. The Rudin–Shapiro construction never exceeds √(2N) at lengths that are powers of two. Erdős conjectured a fixed gap: the maximum is always at least (1 + c) times √N, for some fixed c. Erdős’s original question allowed complex coefficients of size one, and Kahane refuted that version in 1980: those polynomials can be ultraflat, with size close to √N everywhere on the circle. For real signs, the question stayed open. Meanwhile, Balister, Bollobás, Morris, Sahasrabudhe and Tiba proved in 2020 that flat polynomials exist, staying between two fixed multiples of √N. From length two to twenty, our exhaustive search gives best ratios between about 1.15 and 1.43. The same question has a practical form. The merit factor of a sign sequence is N² divided by twice the sum of its squared autocorrelations; it is large when shifted copies of the sequence barely match, which is what radar and communications want. The Barker sequence of length 13 reaches 14.08, random sequences about one, and the best proven families about 6.34. The conjecture that merit factors stay bounded is sometimes called Turyn’s conjecture.

1:49The claims

Result 076 is three OpenAI preprints. The first, 26 pages dated September 23, claims that the best ratio tends to one: for every η greater than zero and every large enough N, some choice of signs keeps the maximum below (1 + η) times √N. That disproves the real-sign form of Erdős’s fixed-gap conjecture. Since the fourth moment is at most the maximum squared times the second, one over the merit factor is at most the squared ratio minus one, so merit factors are unbounded, and Turyn’s conjecture fails. Two later preprints, dated October 5, add a floor. One keeps the size between √N/16 and (1 + η) times √N. The other claims full ultraflatness: between (1 − ε) and (1 + ε) times √N, everywhere on the circle, at every large length. OpenAI published a Lean proof of the first paper’s maximum bound, and of flatness on average, measured with any finite power: 27 files and about 9,600 lines. We have not run it, and the October papers are not formalized.

3:04Chirps and packing

The proof first allows coefficients anywhere between minus one and one. The key tool is a chirp: coefficients whose phase grows quadratically. By stationary phase, a chirp spreads its Fourier mass evenly over an interval whose width is proportional to its curvature. In our example, with curvature one quarter, the spectrum sits at height two, one over the square root of the curvature, on an interval of width one quarter, and nearly vanishes elsewhere. Put chirps on separate blocks of coefficients and give their intervals different centers, and the intervals can tile the circle. In our toy with four blocks and complex coefficients of size one, the size stays within 15% of √N on 98% of the circle. Real coefficients are harder: every interval comes with its mirror image, and the coefficients must stay close to plus or minus one. So the paper samples a real trigonometric polynomial on an auxiliary torus, bounded by one with mean square nearly one, gives each of its modes its own curvature, and packs all the resulting intervals disjointly, using a random hypergraph over a finite field and the Pippenger–Spencer coloring theorem. The last step rounds the relaxed coefficients to signs. A discrepancy argument, the partial coloring method of Beck and Spencer, in the form of Lovett and Meka, changes the polynomial by an amount controlled by how far the coefficients were from signs. Because that defect is small, the change is small compared with √N. In the first October paper, correction waves fill the arcs where the main term is small, and meet the existing waves with matching phases. The ultraflat paper also makes every wave nearly the same height and joins neighboring waves smoothly, so the size stays close to √N everywhere.

4:56Where it stands

None of the three preprints is peer reviewed, and none has been revised. The Lean proof covers the first paper’s maximum bound and its flatness on average; the merit-factor consequence and the two October papers are not formalized. The October papers build on the first. The first paper also derives a consequence in dynamics, and points to result 144 as a related construction. The first paper notes that its result contradicts preprints by el Abdalaoui claiming the opposite, and its appendix points to specific gaps in those arguments. The inputs are published theorems, such as Pippenger and Spencer’s coloring theorem and Lovett and Meka’s discrepancy method. It is not among the papers OSVerify reviewed, and we have seen no expert review. If they hold, polynomials with real sign coefficients can be ultraflat at every large length, like the complex ones Kahane built. Erdős’s fixed-gap conjecture for real signs is false, and so is Turyn’s conjecture on merit factors. The constructions are existential: the papers claim no rate of convergence and no efficient algorithm for finding the signs.