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Claim. The answer to Problem 230 is no. Kahane [Ka80] constructs, for every $n$, a polynomial $P(z)=\sum_k a_kz^k$ of degree $n$ with complex coefficients of modulus one such that

$$\lvert P(z)\rvert=(1+o(1))\sqrt n$$

uniformly on the unit circle, with a relative error that vanishes as the degree grows; such polynomials are called ultraflat. Given any $c>0$, a large $n$ then gives a unimodular polynomial with $\max_{\lvert z\rvert=1}\lvert P(z)\rvert<(1+c)\sqrt n$, so no constant $c>0$ makes the displayed inequality of Problem 230 hold for every $n\ge2$. The site's polynomial carries the coefficients $a_1,\ldots,a_n$; a shift of the index range multiplies $P$ by a power of $z$, which changes nothing on the circle, and one coefficient more or fewer changes the maximum by at most one, which is $o(\sqrt n)$. The lower bound $\sqrt n$ is Parseval's identity, and Hayman's 1974 list [Ha74], where the question is Problem 4.31, attributes the conjecture to Erdős and Newman; Erdős himself expected the answer yes. Littlewood had obtained $(1+o(1))\sqrt n$ away from a small arc, Körner [Ko80] introduced probabilistic ideas for passing from coefficients bounded by one to unimodular ones, and Kahane obtained the uniform statement with relative error $O(n^{-1/17}\sqrt{\log n})$, as the introduction of Bombieri and Bourgain reports (digested on the card bombieri_2009_kahane_ultraflat_polynomials, printed pp. 627--628). Their footnote 1 (p. 627) says that Theorem 2 of Byrnes is incorrect and that its use invalidates the proofs of Körner's Theorems 6 and 7, while his Lemma 2, the basic tool for unimodularity, remains valid. Kahane's paper is not held; the statement above follows the site's commentary and the introduction of Bombieri and Bourgain (pp. 627--628). The sharper construction of Bombieri and Bourgain is a second, independent disproof; the coefficients in both are general points of the unit circle, so neither settles the real-sign question of Problem 1150, which the release's real-sign construction addresses.

Depends on. Nothing in this wiki: the construction is self-contained in Kahane's paper.

Acceptance. Refereed, and credited by the site's curator with the answer no. Reviewed: the site's curator, Thomas Bloom, marks the problem disproved and credits Kahane's ultraflat polynomials in the problem's commentary, which records that Erdős had expected the opposite answer (page last edited 23 January 2026; the proof-claims tab was empty on 2026-10-07); Bombieri and Bourgain state the result as Kahane's theorem in their refereed paper of 2009. Refereed: J.-P. Kahane, Sur les polynômes à coefficients unimodulaires, Bull. London Math. Soc. 12 (1980), no. 5, 321--342, September 1980. Not counted as formalized: the site's label is "DISPROVED (LEAN)", and the statement file of formal-conjectures (FormalConjectures/ErdosProblems/230.lean, added on 2026-09-19 and pinned at that commit; the record link) states erdos_230 as answer(False) under research solved with a formal_proof attribute naming the file Erdos230.lean of Boris Alexeev's lean-proofs repository (added 2026-08-17, pinned at a later commit of that repository; the formalization link). That file declares itself a formalization of a solution to Problem 230 with Kahane as informal author and Codex and GPT-5.6 Sol as formal authors, so it is a formalization link on this page and not a claim of its own; its theorem Erdos230.not_erdos_230 negates the site's question for unimodular coefficients (Lean and Mathlib v4.33.0). This corpus has not built, replayed or audited it.