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Statement

A real Littlewood polynomial of length $N$ is $P(z)=\sum_{k=0}^{N-1}\varepsilon_kz^k$ with every $\varepsilon_k\in\{-1,1\}$; its degree is $N-1$. A family of such polynomials is called ultraflat when $\max_{|z|=1}\bigl||P(z)|/\sqrt N-1\bigr|\to0$ as $N\to\infty$ (p. 1).

Theorem 1. Let $\varepsilon\in(0,1)$. Some integer $N_0$ has the property that each integer $N\ge N_0$ admits signs $\varepsilon_0,\ldots,\varepsilon_{N-1}\in\{-1,1\}$ satisfying

$$(1-\varepsilon)\sqrt N\le\left|\sum_{k=0}^{N-1}\varepsilon_kz^k\right| \le(1+\varepsilon)\sqrt N \qquad\text{for every }|z|=1.$$

The signs may depend on $N$. The two bounds hold for the same polynomial on the entire circle, the real points $z=1$ and $z=-1$ included. The manuscript states no bound on $N_0$ in terms of $\varepsilon$ and no procedure for finding the signs. Problem pages normalize by degree $n$; the statement transfers with $N=n+1$ and $\sqrt{n+1}=(1+o(1))\sqrt n$.

Source. OpenAI, Ultraflat real Littlewood polynomials, release folder preprints/Ultraflat-real-Littlewood-polynomials-October-5-2026; TeX sections/introduction.tex lines 15--29 (label thm:main), PDF p. 1; proof in Section 6, sections/completion.tex, PDF pp. 14--15. The card records the release's attestations and the absence of any refereed, arXiv or independently reviewed version.

Read depth. Claims checked: the statement, the definitions it uses and the two sentences following it were read clause by clause in the TeX source. The proof was read for its structure (below) and no step was checked. Nothing here is independently reviewed.

Proof pointer

Section 6 (pp. 14--15) assembles the theorem from two earlier results. Fix a small $\delta$ and take the function $B_N$ of Proposition 5.1: continuous on the circle, conjugate-symmetric, $1\le|B_N|\le1+C\delta$, real Fourier coefficients with $\sqrt N|\widehat B_N(k)|\le1+C\sqrt\delta$ for $0\le k<N$, and exterior coefficients summing to $O_\delta(N^{-1})$. With $S_\delta=1+C_1\sqrt\delta$ the normalized coefficients $Y_k=\sqrt N\widehat B_N(k)/S_\delta$ lie in $[-1,1]$, and the tail bound makes the projection $U_Y(t)=N^{-1/2}\sum_{k<N}Y_k\mathrm e(kt)$ equal to $B_N/S_\delta+O_\delta(N^{-1})$ uniformly. Parseval and $|B_N|\ge1$ give $N^{-1}\sum Y_k^2\ge S_\delta^{-2}-o(1)$, and since $|Y_k|\ge Y_k^2$ the defect $\mu=\tfrac12\sum(1-|Y_k|)$ satisfies $\mu/N\le q_\delta<1/2$ with $q_\delta=\tfrac12(1-S_\delta^{-2})+\delta\to0$ as $\delta\to0$. This is the hypothesis of Lemma 3.2, which for real inputs returns signs $\varepsilon_k$ with $\|U_\varepsilon-U_Y\|_\infty\le C(N^{-1/2}+\sqrt{q_\delta\log(80/q_\delta)})$. The modulus bounds on $B_N$ then give, uniformly in $t$,

$$S_\delta^{-1}-E_\delta-o(1)\le|U_\varepsilon(t)| \le(1+C\delta)/S_\delta+E_\delta+o(1), \qquad E_\delta=C\sqrt{q_\delta\log(80/q_\delta)};$$

both limits tend to $1$ as $\delta\to0$, so $\delta$ is chosen from $\varepsilon$ and then $N$ taken large. The hypothesis $\varepsilon<1$ is only what makes the lower bound positive; the restriction to large $N$ enters through $N_0(\delta)$ in Proposition 5.1 and the $o(1)$ terms. The closing paragraph notes that no divisibility condition is imposed on $N$ because all auxiliary data (torus dimensions, packing, intervals, leading phases) are fixed before $N$ and the phase identities need nothing about the Fourier index beyond its being an integer.

Dependencies

Internal: Proposition 5.1 (Section 5) and Lemma 3.2 (Section 3), both proved in the manuscript. Through them, two results imported from the companion Nearly minimal maxima and positive minima of Littlewood polynomials without proof: its Lemma 6.2 (real matrix discrepancy, here Lemma 3.1; the companion proves it from Spencer 1985 and Lovett--Meka 2015, Theorem 4 of arXiv:1203.5747v2) and its Lemma 3.1 (signed interval packing, here Lemma 5.2; the companion proves it with Pippenger--Spencer 1989 in the form of Alon--Yuster 2005, Lemma 2.1). Proposition 4.1 adapts the companion's Section 2 construction and is reproved here. Standard inputs: Parseval's identity, the maximum principle and Cauchy's estimate. External premises are taken at statement level; none was checked here.

Bears on

• Problem 1150: the upper bound alone is a claimed negative answer. For $c>0$ take $\varepsilon<c$: the theorem claims degree-$n$ sign polynomials with maximum modulus at most $(1+\varepsilon)\sqrt{n+1}<(1+c)\sqrt n$ for all large $n$, so no constant as the page asks for would exist. The manuscript cites Erdős 1957, Problem 26, and not the catalog number. Unverified here; the page's status rests on acceptance evidence.
• Problem 228: claimed stronger form of the proved statement, both implied constants replaced by $1\mp\varepsilon$ for all large lengths. Unverified here; the page's status rests on the Balister--Bollobás--Morris--Sahasrabudhe--Tiba theorem it records.
• Problem 230: comparison. The page's disproved question allows complex unimodular coefficients; the theorem claims that for each $c>0$ the inequality already fails for real signs at every large length. The manuscript does not name the problem; unverified here, and the page's status rests on the recorded resolution.