Problem 230

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claims/: The 3 claim pages of Problem 230, one per claimant's result; the problem's standing derives from them.

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Statement. Let $P(z)=\sum_{1\leq k\leq n}a_kz^k$ for some $a_k\in \mathbb{C}$ with $\lvert a_k\rvert=1$ for $1\leq k\leq n$. Does there exist a constant $c>0$ such that, for $n\geq 2$, we have

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

Status. DISPROVED (LEAN), the site's label (page last edited 23 January 2026, as of 2026-10-07). The site's curator answers no, against Erdős's own expectation, and credits Kahane [Ka80], whose ultraflat polynomials have $\lvert P(z)\rvert=(1+o(1))\sqrt n$ uniformly on the circle for coefficients of modulus one; the curator names Bombieri and Bourgain [BoBo09] as sharpening the error to $O(n^{7/18}(\log n)^{O(1)})$. The lower bound $\sqrt n$ is Parseval's identity; the site credits Körner [Ko80] with flatness between two constant multiples of $\sqrt n$, but Bombieri and Bourgain (footnote 1, p. 627) record that the proofs of Körner's Theorems 6 and 7 rest on an incorrect theorem of Byrnes; such flatness follows from Kahane's theorem in any case, and two-sided constant-factor flatness for real signs is Problem 228. The question is Problem 4.31 of Hayman's list [Ha74], which attributes the conjecture to Erdős and Newman.

Source. erdosproblems.com/230, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #230, https://www.erdosproblems.com/230.

References.

• [BoBo09] Bombieri, Enrico and Bourgain, Jean, On Kahane's ultraflat polynomials. J. Eur. Math. Soc. (JEMS) (2009), 627-703.
• [Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155-180.
• [Ka80] Kahane, Jean-Pierre, Sur les polynômes à coefficients unimodulaires. Bull. London Math. Soc. (1980), 321-342.
• [Ko80] Körner, T. W., On a polynomial of Byrnes. Bull. London Math. Soc. (1980), 219-224.

Formalization. Statement in formal-conjectures (pinned file, added 2026-09-19): erdos_230 is answer(False) under research solved with a formal_proof attribute naming the file Erdos230.lean of Boris Alexeev's lean-proofs repository, which declares itself a formalization of Kahane's solution with Codex and GPT-5.6 Sol as formal authors (Erdos230.not_erdos_230); this corpus has not built or checked it, and that file is a formalization link on the Kahane claim page, which records the details. The release's declaration OAI.AsymptoticallyMinimalLittlewood.main, built and audited here, is recorded on its claim page.

Current assessment

Disproved; three accepted full claims. The site formulation above (page last edited 2026-01-23) asks for a constant $c>0$ such that every polynomial of degree $n\ge2$ with coefficients of modulus one has maximum modulus at least $(1+c)\sqrt n$ on the unit circle. No such constant exists. Kahane's ultraflat polynomials [Ka80] have $\lvert P(z)\rvert=(1+o(1))\sqrt n$ uniformly on the circle; Bombieri and Bourgain [BoBo09] sharpen the error to $O(n^{7/18+\varepsilon})$ by an effective construction; and the OpenAI release's construction of 2026 reaches $(1+\eta)\sqrt n$ for every $\eta>0$ with coefficients $\pm1$ for every large $n$. The three claim pages, Kahane 1980, Bombieri and Bourgain 2009 and OpenAI 2026, are accepted on different evidence: the first two are refereed papers credited by the site's curator, and the third rests on this corpus's build of the release's Lean declaration and its audit of that declaration's statement, with no outside review recorded. Kahane's paper is not held in the library; its statement follows the site's commentary and the introduction of Bombieri and Bourgain (pp. 627–628). The real-sign form of the question, whether coefficients $\pm1$ force such a constant, is Problem 1150, which the release's construction also answers; the constant-factor flatness of real signs is Problem 228.

Search scope: the site's problem page as exported (last edited 2026-01-23) and its empty proof-claims tab as of 2026-10-07, the formal-conjectures statement file and the header and theorem statement of the lean-proofs file at their pinned commits, the Bombieri–Bourgain paper, and the release's manuscripts and Lean folder at the pinned revision of 2026-10-06; no forum proof claim names this problem.

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.

• erdos_1976_extremal_problems_polynomials
• erdos_1976_extremal_problems_polynomials / problem_p354
• bombieri_2009_kahane_ultraflat_polynomials
• borwein_erdelyi_2003_lower_bounds_merit_factors_trigonometric_polynomials_littlewood_classes
• borwein_erdelyi_2003_lower_bounds_merit_factors_trigonometric_polynomials_littlewood_classes / theorem_3
• hayman_lingham_2018_research_problems_function_theory
• hayman_lingham_2018_research_problems_function_theory / problem_4_13
• hayman_lingham_2018_research_problems_function_theory / problem_4_14
• openai_2026_asymptotically_minimal_maxima_real_littlewood_polynomials
• openai_2026_asymptotically_minimal_maxima_real_littlewood_polynomials / theorem_1_1
• openai_2026_nearly_minimal_maxima_positive_minima_littlewood_polynomials
• openai_2026_nearly_minimal_maxima_positive_minima_littlewood_polynomials / theorem_1_1
• openai_2026_ultraflat_real_littlewood_polynomials
• openai_2026_ultraflat_real_littlewood_polynomials / theorem_1