Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 1150

claims/: The 4 claim pages of Problem 1150, one per claimant's result; the problem's standing derives from them.

Statement. Does there exist a constant $c>0$ such that, for all large $n$ and all polynomials $P$ of degree $n$ with coefficients $\pm 1$ ,

\max_{\lvert z\rvert=1}\lvert P(z)\rvert > (1+c)\sqrt{n}?

Status. OPEN: the site's label (page last edited 23 January 2026; problem page accessed 2026-10-06, its proof-claims tab empty). The derived standing departs from the label, which predates the release: it is solved and disproved, because Theorem 1.1 of the OpenAI release's manuscript of 23 September 2026 gives, for every $\eta>0$ and every large $N$ , signs $\pm1$ whose polynomial of length $N$ has maximum modulus at most $(1+\eta)\sqrt N$ on the circle, so no $c>0$ works; this corpus's verification built its Lean declaration OAI.AsymptoticallyMinimalLittlewood.main with only the three standard axioms and audited its statement, and the corpus accepts it on the claim page. No acknowledgment outside this repository is known. The site's commentary, written before the release, restates the question as whether ultraflat polynomials with coefficients $\pm1$ exist, notes that ultraflat polynomials do exist when the coefficients may be any points of the unit circle (Problem 230), so that the unimodular analogue of the Statement has the answer no, notes that $\max\lvert P\rvert\ge\sqrt n$ is Parseval's identity, and points to the weaker flatness question Problem 228. The problem's discussion thread carries one earlier affirmative claim, el Abdalaoui's preprint of 2025 that $\pm1$ polynomials are never $L^\alpha$ -flat for even $\alpha>2$ , to which the curator and Tao objected and which the accepted theorem contradicts; it has the rejected claim page el Abdalaoui 2025. The same author claimed the affirmative answer earlier, in a 2016 preprint, and again in September 2025; both claims have rejected claim pages (2016, September 2025). Of the two release manuscripts of 5 October 2026, both without Lean, one claims the two-sided ultraflat form $(1-\varepsilon)\sqrt N\le\lvert P(z)\rvert\le(1+\varepsilon)\sqrt N$ and the other the lower bound $\sqrt N/16\le\lvert P(z)\rvert$ with the same upper bound; both are pending on Problem 228's claim page.

Source. erdosproblems.com/1150, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1150, https://www.erdosproblems.com/1150.

Formalization. Statement in formal-conjectures; the file states the problem without a proof and proves only the Parseval lower bound $\max\lvert P\rvert\ge\sqrt{n+1}$ for degree $n$ . The release's declaration OAI.AsymptoticallyMinimalLittlewood.main, built and audited by this corpus's verification against the Statement above, is recorded on the claim page; the statement audit recorded there also compared the declaration with the formal-conjectures statement erdos_1150: its eventual range in $n$ , its coefficient class fixed by the natural degree, and its supremum over the circle against the declaration's pointwise bound.

Current assessment

Disproved by the OpenAI release's construction of 2026-09-23, whose Lean declaration this corpus built and audited (2026-10-07); no acknowledgment outside this repository is known. The question, as the site states it (page last edited 2026-01-23), asks for one $c>0$ such that every $\pm1$ polynomial of every large degree $n$ has maximum modulus above $(1+c)\sqrt n$ on the unit circle; the lower bound $\sqrt n$ is Parseval's identity, and the complex-coefficient relative, where Kahane's ultraflat polynomials give the answer no, is Problem 230. The release's Theorem 1.1 answers no for real signs: the minimal normalized maximum $m_N$ tends to $1$ through all lengths. The accepted claim page records the bridge from length $N$ to degree $N-1$ , the declaration, the build with axioms propext, Classical.choice and Quot.sound only, the comparator pin and the fidelity audit; the evidence is formalized alone, since nobody outside the repository has reviewed or refereed the result and the site lists the problem OPEN with an empty proof-claims tab. What the result trusts is Lean's kernel and the consistency of Mathlib; what it does not give is a lower bound on the circle, as the manuscript's remark after Theorem 1.1 says, nor a convergence rate or a signing algorithm, as the release's family document says.

What remains is quantitative. The release's manuscript notes that Erdélyi's 2026 bound $\max_{\lvert z\rvert=1}\lvert P(z)\rvert^2\ge N+(N-1)^{1/3}/38$ for every $\pm1$ polynomial of length $N$ (card erdelyi_2026_erdos_problem_about_maximum_modulus_littlewood_polynomials_unit_circl) is compatible with the theorem: the excess $\max\lvert P\rvert^2-N$ is unbounded, but $o(N)$ , and its true order between $N^{1/3}$ and $o(N)$ is not determined. The two-sided ultraflat form, $(1-\varepsilon)\sqrt N\le\lvert P(z)\rvert\le(1+\varepsilon)\sqrt N$ on the whole circle for every large $N$ , is claimed by one release manuscript of 2026-10-05, and a second manuscript of the same date claims the lower bound $\sqrt N/16\le\lvert P(z)\rvert$ with the same upper bound; both are without Lean and are claimed on Problem 228's pending claim page; neither is part of this problem's question.

Three claims of an answer by el Abdalaoui are rejected on their claim pages. A 2016 preprint (its page) asserts that no $\pm1$ sequence is $L^4$ -flat, and a September 2025 preprint (its page) asserts the same for every $L^\alpha$ ; Appendix A of the release disputes both. His April 2025 preprint, that $\pm1$ polynomials are never $L^\alpha$ -flat for even $\alpha>2$ , which would give a universal gap, is rejected on its claim page: in the site's thread the curator found that the final step of its proof, on p. 9, does not contradict its Lemma 5, which gives only one function with concentration, and Tao asked that its claims be treated as unconfirmed, and the accepted theorem gives $L^\alpha$ flatness for every finite $\alpha$ along its polynomials, contradicting the conclusion.

A separate release manuscript of 2026-09-23, The circulant Hadamard conjecture, claims that real circulant Hadamard matrices exist only in orders 1 and 4, so that Barker sequences would exist only at lengths 2, 3, 4, 5, 7, 11 and 13, which would make vacuous the Borwein–Mossinghoff consequences of arbitrarily long Barker sequences (card borwein_mossinghoff_2008_barker_sequences_flat_polynomials); its Theorem 1.1, the statement on orders, is formally verified here, as its card records, while of the Barker consequence only the even-length direction is, and the manuscript claims nothing about this problem and gets no claim page.

Search scope: the site's problem page and proof-claims tab (accessed 2026-10-06) and its discussion thread (accessed 2026-10-07), the release's manuscripts, family document and lean/ folder at the pinned revision, the acceptance recorded on the claim page, and the retained cards linked below. The proof-claims tab is empty; the thread's one proof posting, el Abdalaoui's preprint, has the rejected claim page above. No wider literature search was made.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (107)

Problem 228 Ultraflat real Littlewood polynomials Problem 230 Kahane's ultraflat unimodular polynomials Bombieri and Bourgain's quantitative ultraflat polynomials OpenAI's real-sign polynomials with asymptotically minimal maximum el Abdalaoui's square-flatness claim for plus-minus-one polynomials el Abdalaoui's nonflatness claim for plus-minus-one polynomials el Abdalaoui's generalized Littlewood criterion and its sign corollary OpenAI's real-sign polynomials with asymptotically minimal maximum problems/polynomials/E1150/claims Polynomials library/polynomials/balister_2020_flat_littlewood_polynomials_exist library/polynomials/bombieri_2009_kahane_ultraflat_polynomials Barker sequences and flat polynomials The sequence of partial sums of a unimodular power series is not ultraflat Search for Ultraflat Polynomials with Plus and Minus One Coefficients Bonami–Révész: Integral concentration of idempotent trigonometric polynomials Theorem 7: concentration of idempotents on symmetric open sets Theorem 8: concentration of idempotents on symmetric measurable sets Borichev–Sodin–Weiss: Spectra of stationary processes on Z Lemma 3 (p. 8): the delta-prediction lemma Theorem 1 (p. 4): realization spectra lie in the spectrum of the process Theorem 10 (p. 17): Nazarov's Turán-type lemma as the paper restates it Theorem 11 (p. 20): unimodular processes with spectrum in an arc shorter than π are geometric Theorem 3 (p. 7): a finite-valued process with a spectral gap is periodic Theorem 4 (p. 7): Helson's theorem with a period depending only on the values and the spectrum Theorem 5 (p. 11): uniformly discrete values and condition (Θ) give periodic realizations Downarowicz–Lacroix: Merit factors and Morse sequences Corollary 3 (p. 10): singular spectra of continuous binary Morse flows would imply Turyn's conjecture Lemma 0 (p. 2): the merit factor as an L^4 norm Theorem 1 (p. 4): two-sided bounds for the merit factor of a product of words Theorem 2 (p. 9): unbounded merit factors iff a binary Morse flow with square-summable spectral coefficients Extremal problems on polynomials Section 8 problem (pp. 354–355): the Erdős–Newman question on ±1 polynomials Polynomials polynomials/abdalaoui_2025_l_alpha_flatness_erdos_littlewood_s A class of Littlewood polynomials that are not $L^α$-flat Proposition 3.5 (p. 10): L^alpha-flat Littlewood polynomials with 0 < alpha < 2 have frequency of -1 in [1/4, 3/4]Theorem 2.1 (p. 7): Littlewood polynomials whose frequency of -1 lies outside (1/4, 3/4) are not L^alpha-flat Theorem 2.2 (p. 7): unbalanced Littlewood polynomials have L^alpha norms tending to infinity for alpha > 2 Theorem 2.3 (p. 7): palindromic Littlewood polynomials of even degree are not L^alpha-flat polynomials/balister_2020_flat_littlewood_polynomials_exist Theorem 1.1 (p. 1): flat Littlewood polynomials exist in every degree from 2 Theorem 2.1 (p. 3): a centred Littlewood polynomial between 2^-160 sqrt(n) and 2^12 sqrt(n)polynomials/bombieri_2009_kahane_ultraflat_polynomials Borwein–Erdélyi: Lower bounds for merit factors Corollary 2 (p. 3): fourth-moment gap and merit factor bound on the Littlewood class Theorem 1 (p. 2): a fourth-moment gap from an L2 upper bound and a derivative lower bound Theorem 3 (p. 3): conjugate-reciprocal unimodular polynomials have maximum modulus at least sqrt(4/3) sqrt(n+1)

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