Ultraflat Littlewood polynomials and autocorrelation

Source and verification

This article explains family 076 of the October 6, 2026 OpenAI collection. The research conclusions are attributed to its manuscripts. The source review covers statements, introductions, and the proof mechanisms described below; it is not an independent verification of the complete proofs.

Overview

A Littlewood polynomial has every coefficient equal to $1$ or $-1$. Family 076 claims that such polynomials can have almost constant modulus on the entire unit circle, at every sufficiently large length. Its three manuscripts progress from nearly minimal maxima to positive minima and then uniform ultraflatness.

🏷️ Preliminaries: the Parseval scale

Write

$$P(z)=\sum _{j=0}^{N-1}{\epsilon }_j{z}^j,{\epsilon }_j\in \{-1,1\}.$$

Using probability measure $dt$ on $\mathbb{R}/\mathbb{Z}$, orthogonality gives

$${\int }_0^1|P(e^{2\pi it}){|}^{2}dt=N.$$

Thus the maximum modulus is at least $\sqrt{N}$. Ultraflatness means that the whole normalized modulus $|P|/\sqrt{N}$ tends uniformly to one, not merely that the maximum has the right order of magnitude.

The real-coefficient constraint is important. Arbitrary complex numbers of modulus one offer continuous phase freedom; signs offer only two choices per coefficient. Moreover, control at most angles does not ensure control at the real points $z=1$ and $z=-1$.

🏷️ Three increasingly strong claims

Family 076, as stated in the manuscripts

At every sufficiently large integer length, there are real-sign polynomials with maximum at most $(1+\eta )\sqrt{N}$. A companion achieves this upper bound together with minimum at least $\sqrt{N}/16$. The final manuscript claims polynomials $P_N$ satisfying

$$\underset{|z|=1}{\sup }\left|\frac{|P_N(z)|}{\sqrt{N}}-1\right|⟶0.$$

The limit is through all integer lengths. (OpenAI, 2026) (OpenAI, 2026a) (OpenAI, 2026a)

These statements should not be conflated. A nearly optimal upper bound alone can still permit narrow regions where the modulus is small. A fixed positive lower constant gives two-sided flatness but does not force that constant to approach one.

🏷️ A derived consequence: autocorrelation and merit factor

For $1\le u<N$, define the aperiodic autocorrelations

$$C_u=\sum _{j=0}^{N-1-u}{\epsilon }_j{\epsilon }_{j+u}.$$

Expanding the squared modulus gives a constant term $N$ and nonzero Fourier coefficients $C_u$ at frequencies $\pm u$. A second use of Parseval yields

$${\int }_0^1|P(e^{2\pi it}){|}^{4}dt=N^2+2\sum _{u=1}^{N-1}{C}_u^2.$$

The merit factor is $F(P)=N^2/(2\sum C_u^2)$. If $\Vert P{\Vert }_{\infty }\le (1+\eta )\sqrt{N}$, then

$$\begin{array}{c}\int |P{|}^4\\ \le \Vert P{\Vert }_{\infty }^2\int |P{|}^2\\ \le (1+\eta {)}^2{N}^2.\end{array}$$

Consequently $F(P)\ge ((1+\eta {)}^2-1{)}^{-1}$. Thus the first manuscript’s upper-bound claim already implies unbounded merit factor as $\eta \downarrow 0$; ultraflatness is stronger than needed for this consequence.

🏷️ Relaxation followed by controlled rounding

The source initially permits real coefficients $X_j\in [-1,1]$. Define their sign defect by

$$\mu (X)=\frac{1}{2}\sum _j(1-|X_j|).$$

If the mean square of the coefficients is close to one, then the defect is small, since $1-|x|\le 1-x^2$ on $[-1,1]$. The relaxed construction must make the Fourier polynomial both nearly flat and nearly saturated in coefficient energy.

A discrepancy estimate then rounds the coefficients to signs while controlling the polynomial uniformly. In the lower-envelope companion, the normalized error is bounded by an expression of the form

$$\frac{C}{\sqrt{N}}+C\sqrt{\frac{\mu (X)}{N}\log \left(\frac{80N}{\mu (X)}\right)},$$

with the second term interpreted as zero at zero defect. This tends to zero with the relative defect. Independent random rounding would not automatically provide the required uniform control over the entire circle.

🏷️ Oscillatory construction and its refinements

The manuscripts use an auxiliary trigonometric polynomial in several variables and assign its modes to oscillatory waves on short arcs. Stationary phase relates their curvature to the size and position of Fourier coefficients. Packing the arcs and coordinating their phases prevents their leading contributions from adding uncontrollably.

The middle manuscript repairs arcs where the leading amplitude is too small, while maintaining the coefficient constraint and controlling the Fourier tail outside degrees $0,\dots ,N-1$. The final manuscript balances amplitudes more tightly and increases phase curvature near wave endpoints. This suppresses endpoint Fourier contributions without forcing the function’s modulus to become small. Matching phase data across the joins makes the exterior Fourier tail summable.

The order of limits matters: auxiliary geometric parameters are fixed at a chosen accuracy before $N$ tends to infinity. The source’s existence claims should not be read as an efficient signing algorithm or a stated useful rate of convergence.

🏷️ Key steps and proof outlines

Stationary phase spreads one mode over many coefficients

Write $e(t)=e^{2\pi it}$. The basic oscillatory integral is

$$I(u)=\int g(y)e(T\beta y^2/2-uy)dy,$$

where $g$ is a smooth compactly supported amplitude, $T$ is large, and $\beta \ne 0$ is fixed. Completing the square places the stationary point at $y_0=u/(T\beta )$. Fourier inversion then gives the main factor $(T|\beta |{)}^{-1/2}$ times a phase and an amplitude close to $g(y_0)$. Thus one bounded oscillatory wave distributes its Fourier mass among roughly $T|\beta |$ coefficients, each of order $(T|\beta |{)}^{-1/2}$.

The source justifies this uniformly, including when the stationary point approaches or leaves the support. It inserts Gaussian damping, evaluates the resulting Gaussian integral exactly, and removes the damping by dominated convergence against $|\hat{g}|$. Outside the stationary range, integration by parts gives an absolutely summable coefficient tail. Uniformity in $u$ and the tail estimate are essential: a pointwise asymptotic for a fixed coefficient would not control a polynomial’s supremum. (OpenAI, 2026a)

Packing controls interference between modes

The auxiliary real trigonometric polynomial has paired modes $a$ and $-a$, ensuring conjugate symmetry. Each mode is assigned an interval on the circle, together with its reflected interval. The packing lemma selects many compatible collections of intervals without reusing a slot. Its finite combinatorial construction reserves more room than the limiting wave widths need, leaving a fixed margin for smoothing and phase transitions.

Sampling the oscillatory construction at coefficient locations introduces a second issue: sampling can alias distinct frequencies. Poisson summation writes the sampled sum as a sum over integer translates of oscillatory integrals. The packing ensures that the significant stationary contributions occur in the assigned disjoint regions. Nonstationary translates are estimated by integration by parts. This converts geometric separation of wave regions into a uniform bound for the sampled Fourier polynomial.

The resulting coefficients lie in $[-1,1]$ and have nearly full squared energy. Their proximity to signs is measured in aggregate by $\mu (X)$, not by requiring every coefficient to be close to one endpoint. A small set of badly unsaturated coefficients is permitted, provided the total defect is small.

Partial coloring rounds the entire Fourier curve

Choose a sufficiently fine grid on the circle and regard real and imaginary parts of Fourier evaluation as rows of a real matrix. A partial-coloring theorem moves a vector in the coefficient cube until a fixed proportion of its unfixed coordinates reach signs, while controlling all those linear forms. Iteration finishes the rounding. The source refines this by dyadic defect levels, so the error depends on $\mu (X)$ rather than simply on the total number of coefficients.

A derivative bound for a polynomial transfers control from the grid to every point on the circle. This last discretization must have enough points relative to degree; a fixed grid would miss narrow peaks. With $\mu (X)\le c\delta N$, the normalized error becomes

$$O(N^{-1/2})+O\left(\sqrt{\delta \log (1/\delta )}\right).$$

First fix $\delta$ and all packing data, then let $N\to \infty$, and only then let $\delta \downarrow 0$. This order gives an all-length asymptotic statement without claiming a quantitative rate.

Repairing zeros and then balancing every amplitude

The positive-minimum companion adds a correction on regions where the original waves have small modulus. Its phase derivatives are assigned so that Fourier coefficients remain small, and its phase is matched where the correction turns on. A slight global rescaling restores the coefficient constraint $[-1,1]$. The proof budgets three errors separately: the correction’s Fourier tail, the coefficient rescaling, and the final rounding. Choosing them below the available lower-envelope margin leaves the stated positive minimum. (OpenAI, 2026b)

The ultraflat companion requires the stronger condition that all leading wave amplitudes approach one. It constructs balanced auxiliary coefficients and varies inverse curvature in the packed intervals to synchronize stationary-phase amplitudes. Enhanced endpoint curvature suppresses endpoint contributions while preserving modulus. Projection to the allowed coefficient range and defect-sensitive rounding then leave a uniformly small error. These extra amplitude and endpoint arguments explain why the maximum-only construction cannot by itself prove ultraflatness. (OpenAI, 2026c)

See Also

• Family 075: Fourier convergence concerns reconstructing arbitrary functions from coefficients; here the coefficients are designed to impose a prescribed modulus.
• on l2 decoupling theorem develops another setting where interactions among separated frequency pieces determine a global norm.

🏷️ Manuscripts and verification scope

• Ultraflat real Littlewood polynomials — the pinned manuscript; its directory supplies the source and citation metadata.
• Nearly minimal maxima and positive minima of Littlewood polynomials — the pinned manuscript; its directory supplies the source and citation metadata.
• Asymptotically minimal maxima of real Littlewood polynomials — the pinned manuscript; its directory supplies the source and citation metadata.

The repository provides a formalization-scope page. Despite its broad heading, the scope and comparator list describe asymptotically minimal maxima and flatness in every fixed finite Lᵖ mean. They do not document the final manuscript’s uniform two-sided ultraflatness claim. These are statements about the repository’s documentation: no Lean build, axiom audit, or comparator-equivalence check was performed for this article.

OpenAI collection gives the reading sequence and source policy; coverage register records which families have articles and which remain to be written.

🐻  OpenAI 2026a. Asymptotically minimal maxima of real Littlewood polynomials.

🐻  OpenAI 2026b. Nearly minimal maxima and positive minima of Littlewood polynomials.

🐻  OpenAI 2026c. Ultraflat real Littlewood polynomials.