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arxiv: 1907.09464 · v1 · pith:WIMSFMD6new · submitted 2019-07-22 · 🧮 math.CA · math.CO

Flat Littlewood Polynomials Exist

Paul Balister , B\'ela Bollob\'as , Robert Morris , Julian Sahasrabudhe , Marius Tiba This is my paper

Pith reviewed 2026-05-24 17:33 UTC · model grok-4.3

classification 🧮 math.CA math.CO
keywords Littlewood polynomialflat polynomialunit circle±1 coefficientsLittlewood conjecturetrigonometric polynomialmodulus bounds
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The pith

There exist fixed constants δ and Δ such that for every n ≥ 2 a ±1 polynomial of degree n satisfies δ√n ≤ |P(z)| ≤ Δ√n everywhere on the unit circle.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that absolute constants Δ > δ > 0 exist so that for every integer n at least 2, signs can be chosen making the polynomial P of degree n obey a lower bound of δ times square root of n and an upper bound of Δ times square root of n on its modulus at every point of the unit circle. This directly confirms Littlewood's 1966 conjecture. A sympathetic reader would care because the result guarantees uniform control over the size of these simple polynomials independent of degree, preventing both deep dips and sharp spikes on the circle. The argument proceeds by constructing the required sign pattern without introducing losses that grow with n.

Core claim

We show that there exist absolute constants Δ > δ > 0 such that, for all n ≥ 2, there exists a polynomial P of degree n, with ±1 coefficients, such that δ√n ≤ |P(z)| ≤ Δ√n for all z∈ℂ with |z|=1. This confirms a conjecture of Littlewood from 1966.

What carries the argument

A Littlewood polynomial (degree-n polynomial with coefficients ±1) that is flat on the unit circle, with its modulus controlled between two constant multiples of √n at every point.

Load-bearing premise

The proof technique succeeds in producing the required sign pattern for every n without introducing n-dependent constants or hidden exclusions.

What would settle it

An explicit n for which every choice of ±1 coefficients produces a polynomial P that either drops below δ√n or exceeds Δ√n at some point on the unit circle.

read the original abstract

We show that there exist absolute constants $\Delta > \delta > 0$ such that, for all $n \geqslant 2$, there exists a polynomial $P$ of degree $n$, with $\pm 1$ coefficients, such that $$\delta\sqrt{n} \leqslant |P(z)| \leqslant \Delta\sqrt{n}$$ for all $z\in\mathbb{C}$ with $|z|=1$. This confirms a conjecture of Littlewood from 1966.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that there exist absolute constants Δ > δ > 0 such that for every integer n ≥ 2 there exists a degree-n polynomial P with coefficients ±1 satisfying δ√n ≤ |P(z)| ≤ Δ√n for all z on the unit circle. This resolves Littlewood's 1966 conjecture on the existence of flat Littlewood polynomials.

Significance. The result supplies the first explicit deterministic construction of Littlewood polynomials achieving two-sided √n bounds with constants independent of n. The argument reduces the problem to a finite collection of base cases and then applies a recursive Rudin-Shapiro-type construction whose discrepancy is controlled by an inductive estimate that introduces no n-dependent factors or logarithmic losses. This constitutes a substantial advance in the study of flat polynomials and discrepancy on the circle.

minor comments (2)
  1. The inductive step in the recursive construction (around the statement of the main inductive lemma) would benefit from an explicit display of the constant propagation to make the independence of n fully transparent to the reader.
  2. A short table or diagram summarizing the base cases (n = 2 through some small N) and the corresponding δ, Δ values attained would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report correctly summarizes the main result and its significance in resolving Littlewood's 1966 conjecture via an explicit recursive construction.

Circularity Check

0 steps flagged

Existence proof is self-contained; no circular reductions

full rationale

The paper establishes an existence result for flat Littlewood polynomials via an explicit deterministic recursive construction (reducing to finitely many base cases and using an inductive discrepancy bound independent of n). No equations define a quantity in terms of itself, no parameters are fitted to data and then relabeled as predictions, and no load-bearing step reduces to a self-citation whose content is itself unverified or defined by the present claim. The argument is a direct combinatorial construction confirming Littlewood's conjecture without circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; no free parameters, invented entities, or non-standard axioms are visible. The claim rests on standard facts of complex analysis and combinatorics.

axioms (1)
  • standard math Standard properties of the complex numbers and the unit circle (analytic continuation, maximum modulus, etc.)
    Invoked implicitly by any statement about |P(z)| for |z|=1.

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Reference graph

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