Real ultraflat Littlewood polynomials and unbounded binary merit factors

1 · The essentials

Start with a polynomial whose coefficients are all either $+ 1$ or $- 1$ . The main manuscript claims that these signs can be chosen so that the polynomial’s size stays almost equal to $N$ as its input travels around the entire unit circle, where $N$ is the number of coefficients. The claim includes every point on the circle, even the real points $1$ and $- 1$ . Any prescribed relative error is allowed, provided the length is sufficiently large. This works at every sufficiently large integer length, although the signs can be chosen afresh at each length. A companion result has a Lean formalization covering the smallest achievable maximum size and a consequence for binary merit factors, rather than explicitly covering the full two-sided claim.

A picture to help (an imperfect image, not to be taken literally)

As an imperfect image, imagine choosing a row of plus and minus settings to keep a dial reading almost constant throughout one complete turn. The full claim controls both upward and downward deviations; controlling only the highest reading is a weaker requirement.

Why it matters

The family distinguishes between making a polynomial’s largest size nearly minimal and keeping its size nearly constant everywhere. The Lean-formalized companion addresses the former and states that the largest binary merit factor tends to infinity through all integer lengths. It identifies these conclusions as contradicting the real-sign analogue of Erdős’s fixed relative-gap conjecture and Turyn’s bounded-merit-factor conjecture.

2 · Going further

A real Littlewood polynomial here has the form $P_{N} (z) = \sum_{k = 0}^{N - 1} a_{k} z^{k}$ , with $a_{k} \in {- 1, 1}$ . The main manuscript claims that for every $ε \in (0, 1)$ and every sufficiently large integer $N$ , one can choose the coefficients so that $(1 - ε) N \leq ∣ P_{N} (z) ∣ \leq (1 + ε) N$ for every $∣ z ∣ = 1$ . This is a uniform, two-sided assertion: no point of the circle is excluded, including $z = 1$ and $z = - 1$ . The choices need not form one consistent sequence of signs across lengths.

The second manuscript gives an intermediate statement. For every $η > 0$ and every sufficiently large integer $N$ , it claims a choice satisfying $N /16 \leq ∣ P_{N} (z) ∣ \leq (1 + η) N$ throughout the circle. Its upper bound approaches the same target, but its lower bound remains a fixed fraction of that target rather than approaching it.

The Lean-formalized third manuscript concerns $M_{N} = min_{a_{0}, \dots, a_{N - 1} \in {- 1, 1}} max_{∣ z ∣ = 1} ∣ P_{N} (z) ∣$ and establishes $M_{N} = (1 + o (1)) N$ as $N$ tends to infinity through all integers. Its abstract also states that the largest binary merit factor at length $N$ tends to infinity, and identifies the two conclusions as refuting the corresponding conjectures of Erdős and Turyn. The supplied abstracts do not describe a construction or the steps of the argument, so they support this comparison of results but not an explanation of the proof mechanism.

Useful notions

Polynomials and their coefficients
Complex numbers and the unit circle
Modulus as the size of a complex number
Uniform bounds
Limits and relative error

What was known before / what is new

Before

The source refers to a real-sign analogue of Erdős’s fixed relative-gap conjecture about the smallest possible maximum modulus. It also refers to Turyn’s conjecture that binary merit factors remain bounded. It supplies no further historical background or earlier quantitative bounds.

New

The main manuscript claims two-sided ultraflatness for real Littlewood polynomials at every sufficiently large integer length, including control at the real endpoints. The second manuscript claims nearly minimal maxima together with a lower bound of $N /16$ . The Lean-formalized companion establishes asymptotically minimal maxima and states the resulting divergence of the largest binary merit factors through all integer lengths.

Limits and assumptions

The coefficients must be real signs, and the assertions concern polynomials with consecutive coefficients and sufficiently large lengths. The required length may depend on the prescribed error; the supplied abstracts give no explicit threshold. Signs may be chosen separately at each length, so no compatible choice across lengths is asserted. The family has a Lean formalization attached to Manuscript 3, whose stated scope concerns asymptotically minimal maxima and the merit-factor consequence; the supplied material does not identify the stronger two-sided bounds in Manuscripts 1 and 2 as formalized. No further remaining open questions are specified.

Mini glossary

Real Littlewood polynomial: A polynomial whose consecutive coefficients are all either $- 1$ or $1$ .
Unit circle: The set of complex numbers $z$ satisfying $∣ z ∣ = 1$ .
Modulus: The size or absolute value of a complex number.
Ultraflatness: Here, the property that a length- $N$ polynomial’s modulus divided by $N$ approaches $1$ uniformly over the entire unit circle.
$o (1)$: A quantity that tends to zero as the length $N$ tends to infinity.

Keywords : Littlewood polynomials · Ultraflatness · Binary merit factors

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Claim made in the manuscript (source text)

Constructs polynomials with N consecutive coefficients in ${- 1, 1}$ whose modulus is $(1 + o (1)) N$ uniformly on the entire unit circle, for every sufficiently large integer length N. Thus real Littlewood polynomials are ultraflat, including at the real endpoints. Their binary merit factors tend to infinity, disproving Turyn's bounded-merit-factor conjecture.

Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.

Scope of the Lean formalization (source text)

…

lean/docs/076.md

Manuscripts in this family

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Ultraflat real Littlewood polynomials

05/10/2026

To be verified

For every $ε \in (0, 1)$ and every sufficiently large integer N, there is a polynomial of length N with coefficients in ${- 1, 1}$ whose modulus lies between $(1 - ε) N$ and $(1 + ε) N$ everywhere on the unit circle. Thus real Littlewood polynomials can be ultraflat through every sufficiently large integer length. The signs may be chosen separately at each length.

PDF GitHub folder

Nearly minimal maxima and positive minima of Littlewood polynomials

05/10/2026

To be verified

For every η > 0 and every sufficiently large integer N, there is a polynomial with N consecutive coefficients in ${- 1, 1}$ whose modulus lies between $N /16$ and $(1 + η) N$ everywhere on the unit circle.

PDF GitHub folder

Asymptotically minimal maxima of real Littlewood polynomials

23/09/2026

Verified by Lean

We prove that the minimum possible maximum modulus on the unit circle of a polynomial with N consecutive real coefficients in ${- 1, 1}$ is $(1 + o (1)) N$ , as N tends to infinity through all integers. This disproves the real-sign analogue of Erdős's fixed relative-gap conjecture. As a consequence, the largest binary merit factor at length N tends to infinity through all integer lengths, disproving Turyn's conjecture.

PDF GitHub folder Lean formalization