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

Updated Oct 8

Problem 228

claims/: The 2 claim pages of Problem 228, one per claimant's result; the problem's standing derives from them.

Statement. Does there exist, for all large $n$ , a polynomial $P$ of degree $n$ , with coefficients $\pm 1$ , such that

\sqrt{n} \ll \lvert P(z) \rvert \ll \sqrt{n}

for all $\lvert z\rvert =1$ , with the implied constants independent of $z$ and $n$ ?

Status. PROVED (LEAN), the site's label (page last edited 2026-01-23). The Lean marker reflects the file Erdos228.lean of Boris Alexeev's lean-proofs repository, which declares itself a formalization of the theorem of Balister, Bollobás, Morris, Sahasrabudhe and Tiba with Codex and GPT-5.6 Sol as formal authors; it is a formalization link on their claim page, and nothing was built or audited here. The formal-conjectures file under Formalization states the theorem and carries no proof. The answer is yes for every $n\ge2$ : Balister, Bollobás, Morris, Sahasrabudhe and Tiba [BBMST20] construct the polynomial with absolute constants, refereed in the Annals of Mathematics and credited by the site, which the corpus accepts on the claim page [[problems/polynomials/E0228/claims/2019_07_22_balister_bollobas_morris_sahasrabudhe_tiba|Balister et al. 2020]]. A stronger form, with the ratio of $\lvert P(z)\rvert$ to $\sqrt n$ forced to $1$ uniformly on the circle, is claimed by the OpenAI release of October 2026 and stays pending on OpenAI 2026.

Source. erdosproblems.com/228, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #228, https://www.erdosproblems.com/228.

References.

[BBMST20] Balister, Paul and Bollobás, Béla and Morris, Robert and Sahasrabudhe, Julian and Tiba, Marius, Flat Littlewood polynomials exist. Ann. of Math. (2) 192 (2020), no. 3, 977–1004.

Formalization. Statement in formal-conjectures, pinned to the file's last change (2026-09-18); its theorem erdos_228 is tagged solved there, its proof is sorry and it carries no formal_proof attribute, so the file is a statement of the problem and not a formalization of the answer. The Lean the site's label refers to is the file Erdos228.lean of Boris Alexeev's lean-proofs repository (added 2026-08-21, last changed 2026-08-23), which declares itself a formalization of a solution to the problem with Balister, Bollobás, Morris, Sahasrabudhe and Tiba as informal authors and Codex and GPT-5.6 Sol as formal authors and proves Erdos228.erdos_228, the formal-conjectures statement; the community database records a Lean formal status for the problem (entry last updated 2026-08-23). Nothing was built or checked here, and that file is a formalization link on the Balister et al. claim page, which records the details.

Current assessment

The question, as the site states it (page last edited 2026-01-23), asks for one $\pm1$ polynomial of each large degree $n$ whose modulus is bounded above and below by fixed multiples of $\sqrt n$ on the whole unit circle. Erdős asked it in 1957 as his Problem 26 and Littlewood conjectured the answer in 1966; the site traces the conjecture itself to Littlewood. The upper bound was classical, from the Rudin–Shapiro polynomials; the lower bound, long known only in the form $n^{0.431}$ , is what [BBMST20] proved, for every $n\ge2$ rather than only for large $n$ . That paper is the accepted answer: refereed (Ann. of Math. (2) 192 (2020), 977–1004) and credited by the site, with its construction and constants digested on the card [[../library/polynomials/balister_2020_flat_littlewood_polynomials_exist/_index|Balister et al. 2020]]. The project has not verified the proof itself.

Pending beside it is the OpenAI release's claim of October 2026 that the constants can be taken as $1-\varepsilon$ and $1+\varepsilon$ for every large degree, so that the modulus is asymptotically $\sqrt n$ everywhere on the circle, with the same family claimed to answer Problem 1150 negatively. The release's Lean states only the upper bound and a finite-exponent flatness, not the lower bound the question needs, so the claim rests on manuscripts and stays claimed on [[problems/polynomials/E0228/claims/2026_10_05_openai|its page]]; it does not change this problem's standing, which the accepted answer already fixes. The complex-coefficient relative, where unimodular ultraflat polynomials exist by Kahane's theorem, is Problem 230.

Search scope: the site's problem page as exported (last edited 2026-01-23), the community database's entry (Lean formal status dated 2026-08-23), the header and theorem statement of the lean-proofs file Erdos228.lean at its pinned commit, the [BBMST20] paper (arXiv:1907.09464) and the release's manuscripts and Lean folder at the pinned revision of 2026-10-06; no forum proof claim names this problem. No wider literature search was made, none being needed for a refereed answer the site credits.

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 (28)

Flat Littlewood polynomials exist Ultraflat real Littlewood polynomials problems/polynomials/E0228/claims Problem 230 OpenAI's real-sign polynomials with asymptotically minimal maximum Problem 1150 Polynomials library/polynomials/balister_2020_flat_littlewood_polynomials_exist A class of Littlewood polynomials that are not $L^α$-flat Theorem 2.2 (p. 7): unbalanced Littlewood polynomials have L^alpha norms tending to infinity for alpha > 2 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)Theorem 2.3 (p. 5): a cosine polynomial small only on few short separated intervals Theorem 2.4 (p. 6): an odd-frequency sine polynomial large on prescribed intervals Barker sequences and flat polynomials Theorem 3.1: a Barker polynomial of length n satisfies (alpha_1 + O(1/n)) sqrt n <= |f| <= (alpha_2 + O(1/n)) sqrt n on the circle polynomials/hayman_lingham_2018_research_problems_function_theory Problem 4.14: a plus-minus-one polynomial bounded below by a multiple of sqrt(n) on the unit circle Search for Ultraflat Polynomials with Plus and Minus One Coefficients Conjecture, p. 4: the limits of M_n, m_n and W_n exist, with M about 1.27, m about 0.64 and W about 0.79 Conjecture, p. 9, inequality (16): for all large n some plus or minus one polynomial has 0.5 < |F(z)|/sqrt(n+1) < 1.5 on the circle polynomials/openai_2026_asymptotically_minimal_maxima_real_littlewood_polynomials Theorem 1.1: the minimal normalized maximum of real Littlewood polynomials tends to one polynomials/openai_2026_nearly_minimal_maxima_positive_minima_littlewood_polynomials Theorem 1.1: ±1 polynomials of every large length N with modulus between √N/16 and (1+η)√N on the circle polynomials/openai_2026_ultraflat_real_littlewood_polynomials Theorem 1: sign polynomials of every large length N with modulus within a factor 1 ± ε of √N on the whole circle

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