Nearly constant waves made only from plus and minus signs

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Family 076 · Real and complex analysis
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-14

1. The question

Can a sum of waves, each carrying only a plus or minus sign, have almost the same amplitude at every point of a circle?

With arbitrary complex coefficients, the phases can be adjusted continuously. Restricting every coefficient to $+1$ or $-1$ removes most of that freedom. Yet the question still asks for uniform control at every angle, including places where many waves might reinforce or cancel one another.

The three manuscripts in family 076 describe a progression. The first chronological paper claims an asymptotically smallest possible maximum amplitude. A refinement also keeps the minimum away from zero. The strongest paper claims that both extrema approach the same natural value, giving real ultraflat Littlewood polynomials at every sufficiently large integer length. The family also states consequences for binary sequences with small autocorrelations.

2. Background and essential definitions

The unavoidable square-root scale

A real Littlewood polynomial of length $N$ is

$$P(z)=\sum_{k=0}^{N-1}\varepsilon_k z^k, \qquad \varepsilon_k\in\{-1,1\}.$$

Length counts the coefficients, so the degree is $N-1$. On the unit circle, write $z=e^{2\pi it}$, where $0\le t<1$. Orthogonality of the exponential waves gives Parseval’s identity:

$$\int_0^1|P(e^{2\pi it})|^2\,dt=N.$$

The root-mean-square amplitude is therefore exactly $\sqrt N$. In particular, the maximum cannot be smaller than $\sqrt N$. The challenge is whether the maximum can approach that value without large exceptional peaks or deep troughs.

Constant-factor flatness means keeping the modulus between $c\sqrt N$ and $C\sqrt N$, for fixed positive constants. Ultraflatness asks for the stronger condition

$$\sup_{|z|=1}\left|\frac{|P(z)|}{\sqrt N}-1\right|\longrightarrow0.$$

The coefficients and even the entire polynomial may change with $N$. This definition does not require successive polynomials to be partial sums of one infinite sign sequence.

Binary correlations and merit factor

A binary word $A=(a_0,\ldots,a_{N-1})$ with entries in $\{-1,1\}$ has aperiodic autocorrelations

$$C_u(A)=\sum_{j=0}^{N-1-u}a_j a_{j+u},\qquad 1\le u<N.$$

These compare the word with a shifted copy over their overlapping positions. “Aperiodic” means that entries are not wrapped around the ends.

Its merit factor is

$$F(A)=\frac{N^2}{2\sum_{u=1}^{N-1}C_u(A)^2}.$$

A large value means that the total squared correlation away from zero shift is small. Such questions arise when one wants a signal to resemble itself strongly only at its correct alignment.

For $P_A(z)=\sum a_jz^j$, orthogonality also gives

$$\int_0^1|P_A(e^{2\pi it})|^4\,dt =N^2+2\sum_{u=1}^{N-1}C_u(A)^2.$$

Thus flattening the polynomial’s modulus suppresses the correlation energy. This identity is the bridge between the analytic and binary-sequence statements.

3. Development history and previous partial results

Littlewood and Erdős posed several related flatness problems with different coefficient constraints. It is essential to distinguish real signs from arbitrary complex coefficients of modulus one.

The Rudin–Shapiro construction produces real-sign polynomials whose maximum is at most $\sqrt{2N}$ at dyadic lengths. This gets the correct order of growth, but leaves a fixed multiplicative gap above the Parseval minimum.

Kahane constructed ultraflat polynomials with complex unimodular coefficients in 1980. Bombieri and Bourgain later supplied quantitative refinements. Those coefficients can point anywhere on the unit circle; replacing them by real signs is an additional problem.

Balister, Bollobás, Morris, Sahasrabudhe and Tiba’s Flat Littlewood polynomials exist, published in the Annals of Mathematics in 2020, established two-sided constant-factor flatness for real signs. Their construction controlled both extrema at once, but did not make both constants converge to one.

On the sequence side, Jedwab, Katz and Schmidt’s 2013 paper Littlewood polynomials with small L4 norm constructed families with limiting merit factor approximately $6.342061$. The conjecture commonly attributed to Turyn predicts a universal bound on binary merit factors. The new family claims that no such bound exists.

Some recent obstructions concern stronger or different quantifiers. Erdélyi’s lower bound on the maximum modulus permits an additive gap above $N$ in the squared maximum while its relative size tends to zero. Another obstruction discussed in the strongest manuscript concerns all partial sums of one fixed power series. Neither statement has the same form as choosing a new sign pattern independently at each length.

4. What the new manuscripts claim

Three levels of flatness

The September manuscript states that the best normalized maximum tends to one:

$$\min_{\varepsilon_k\in\{-1,1\}} \frac1{\sqrt N}\max_{|z|=1} \left|\sum_{k=0}^{N-1}\varepsilon_kz^k\right| \longrightarrow1.$$

This holds through all integer lengths. By itself it supplies no uniform positive lower bound for the same polynomial.

The first October refinement states that, for every $\eta>0$ and every sufficiently large $N$, one choice of signs gives

$$\frac{\sqrt N}{16}\le |P(z)|\le(1+\eta)\sqrt N \qquad(|z|=1).$$

The strongest October manuscript replaces the fixed lower constant by an asymptotically optimal one. For every $0<\varepsilon<1$, all sufficiently large $N$ admit signs satisfying

$$(1-\varepsilon)\sqrt N\le|P(z)|\le(1+\varepsilon)\sqrt N \qquad\text{for every }|z|=1.$$

The same polynomial satisfies both bounds, including at $z=1$ and $z=-1$. The signs may be chosen separately for different lengths.

Correlation consequences

Already the asymptotically minimal maximum implies that the largest merit factor among binary words of length $N$ tends to infinity as $N\to\infty$. The fourth-moment identity explains why: a squared modulus with mean $N$ and maximum nearly $N$ has fourth moment nearly $N^2$, leaving a vanishing relative contribution from nonzero autocorrelations.

The September paper further applies a theorem of Downarowicz and Lacroix to obtain a symbolic dynamical system with a specified absolutely continuous spectral measure. This is a consequence through an existing theorem, not an assertion that every binary sequence has such spectral behavior.

5. Why the advance matters

Importance: 5/5. If correct, real ultraflatness settles a central restricted-coefficient problem and overturns a long-standing boundedness prediction for binary merit factors. The claim controls every angle and every sufficiently large length, which is substantially stronger than average flatness or a construction along a sparse subsequence.

The family also illustrates a powerful connection between continuous analysis and discrete design. The modulus of a complex polynomial encodes many correlations of a finite sign sequence simultaneously. Improving a uniform analytic bound can therefore force a qualitative change in what binary patterns can achieve.

The significance is asymptotic and structural. The manuscripts do not provide a practical signing algorithm or numerical length at which a prescribed degree of flatness becomes available, so immediate engineering performance is not part of the claim.

6. The main idea

First permit almost-sign coefficients

Directly choosing $N$ signs while controlling infinitely many angles is difficult. The construction first allows coefficients $X_k\in[-1,1]$. These continuous choices can be shaped using oscillatory analysis.

If their mean-square mass is nearly maximal, most of their total coefficient mass must be close to sign size. The deficit can be recorded by

$$D(X)=\frac12\sum_{k=0}^{N-1}(1-|X_k|).$$

A discrepancy estimate then rounds the coefficients to signs while controlling the change in the polynomial uniformly on the circle. Small total deficit makes the rounding error small relative to $\sqrt N$. Independent rounding would not supply the same coordinated control over all angles.

Spread and coordinate the waves

The manuscripts build auxiliary trigonometric polynomials on a higher-dimensional torus. Quadratic phases spread their Fourier contributions across assigned intervals. A packing argument arranges these contributions so that the largest terms do not all reinforce one another at the same angle.

Stationary phase is the analytical guide: an oscillatory integral is largest near points where its phase derivative matches the requested frequency. Choosing the curvature and placement of each wave controls where its Fourier contribution appears and how large it is.

The first construction obtains nearly full mean-square mass with an almost minimal maximum, but some arcs can still have small amplitude. The intermediate paper repairs those arcs with phase-matched corrections whose coefficient cost stays small. This secures a fixed lower bound before rounding.

Keep the minimum close to the maximum

The ultraflat paper strengthens the wave construction itself. Its waves have almost constant modulus, and their phases coordinate the main Fourier contributions through the auxiliary polynomial. Near wave endpoints, increased phase curvature suppresses unwanted Fourier contributions without lowering the wave amplitude.

Joining the waves requires matching values and leading phase derivatives across the gaps. The described matching cancels leading boundary terms and keeps the Fourier mass outside the permitted degrees small. Projection to the desired coefficient interval and the final rounding can then preserve both upper and lower bounds.

All auxiliary choices are fixed before the final length tends to infinity. That order of parameters is how the manuscripts obtain every sufficiently large length rather than only a selected sequence of lengths.

7. Scope, limitations, and open questions

The strongest theorem chooses a separate polynomial at each length. It does not produce one infinite sign sequence whose every sufficiently long initial segment is ultraflat.

Uniform modulus does not mean the polynomial itself is nearly constant: its complex argument can vary substantially around the circle. Nor is exact constant modulus asserted for a finite nontrivial Littlewood polynomial.

The first paper’s upper bound, the second paper’s fixed lower bound and the final ultraflat claim should remain distinct. A small maximum plus Parseval’s identity gives strong averaged flatness, but does not alone rule out narrow deep troughs. The later lower-bound constructions address that extra difficulty.

The sequence consequence concerns the stated aperiodic merit-factor normalization. It does not automatically transfer to periodic correlations or other signal-quality objectives. Effective rates, practical constructions and the dependence of the threshold length on the accuracy remain separate quantitative questions.

8. Sources and further reading

The three source manuscripts below preserve the progression from maximum control to a positive minimum and then ultraflatness. Primary historical references include Balister, Bollobás, Morris, Sahasrabudhe and Tiba, Flat Littlewood polynomials exist (2020); Kahane, Sur les polynômes à coefficients unimodulaires (1980); and Jedwab, Katz and Schmidt, Littlewood polynomials with small L4 norm (2013).

Downarowicz and Lacroix’s Merit factors and Morse sequences provides the cited spectral connection. Optional background on Littlewood polynomials and autocorrelation explains the two sides of the problem.

Source manuscripts

• M1. OpenAI, Ultraflat real Littlewood polynomials (October 5, 2026).
• M2. OpenAI, Nearly minimal maxima and positive minima of Littlewood polynomials (October 5, 2026).
• M3. OpenAI, Asymptotically minimal maxima of real Littlewood polynomials (September 23, 2026).

Sources consulted on 14 October 2026 at repository revision 1581d4c. Download the manuscript citations.