Ultraflat $\pm 1$ (Littlewood) polynomials: must $\max_{|z|=1}|P(z)|>(1+c)\sqrt{n}$? (Erdős #1150)
Statement
Call $P(z)=\sum_{k=0}^{n}\epsilon_k z^k$ with every $\epsilon_k\in\{-1,+1\}$ a Littlewood polynomial. Does there exist a constant $c>0$ such that, for all large $n$ and all Littlewood polynomials $P$ of degree $n$, $$\max_{\lvert z\rvert=1}\lvert P(z)\rvert > (1+c)\sqrt{n}?$$ By Parseval, $\max_{\lvert z\rvert=1}\lvert P(z)\rvert\geq \lVert P\rVert_{L^2(\lvert z\rvert=1)}=\sqrt{n+1}$ is trivial; the question is whether the sup-norm must beat this $L^2$ benchmark by a fixed factor — equivalently, whether 'ultraflat' Littlewood polynomials, i.e. a sequence with $\max_{\lvert z\rvert=1}\lvert P_{n_j}(z)\rvert=(1+o(1))\sqrt{n_j}$, do NOT exist.
Acceptance. FULLY RESOLVES: a proof that some explicit $c>0$ works for all sufficiently large $n$ — machine-checkable (Lean, e.g. against the formal-conjectures statement) preferred, else a complete written proof; if the argument leans on the claimed $L^\alpha$-flatness result cited in the background, that result's proof must be independently verified or reproduced in full, not merely cited. OR a disproof: an explicit construction (a computable rule producing $\pm 1$ coefficient sequences for an infinite set of degrees) together with a proof that $\max_{\lvert z\rvert=1}\lvert P_n\rvert = (1+o(1))\sqrt{n}$ along the sequence — certified numerics on finitely many degrees alone cannot establish the asymptotic and must accompany, not replace, the proof. ADVANCES: (a) certified exact values (or rigorous two-sided enclosures) of $m(n)=\min_P\max_{\lvert z\rvert=1}\lvert P\rvert$ over all $2^{n+1}$ Littlewood polynomials for a verified range of $n$ beyond what is documented in prior literature, with search code and an exhaustiveness certificate; (b) a proof of the conjecture for a structured subclass (e.g. skewsymmetric or self-reciprocal Littlewood polynomials); (c) a verified confirmation or refutation of a specific load-bearing step of the arXiv:2504.21499 claim, written up reproducibly. Deliver the proof file, or the search code plus the certified $m(n)$ table.
Background
A well-known conjecture of Erdős (he conjectured ultraflat $\pm 1$ polynomials do not exist), recorded as Problem 4.31 in Hayman's 'Research Problems in Function Theory' [Ha74] and as [Va99, 2.36]; listed as open on erdosproblems.com/1150 (fetched 2026-07-13, status 'open', tagged 'analysis | polynomials'). Contrast 1: if the coefficients may be arbitrary unimodular complex numbers, ultraflat polynomials DO exist (Kahane's theorem — see Erdős #230, erdosproblems.com/230), so the $\pm 1$ restriction is essential. Contrast 2: the weaker flatness question — Littlewood's conjecture that there are $\pm 1$ polynomials with $\delta\sqrt{n}\le \lvert P(z)\rvert\le \Delta\sqrt{n}$ on the whole circle — is Erdős #228 and was resolved affirmatively by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba (2019), so flat is achievable while ultraflat is conjectured impossible. Closely related is the Erdős–Newman merit-factor circle of problems ($L^4$ analogue). Caution: a 2025 preprint of el Abdalaoui (arXiv:2504.21499) claims Littlewood polynomials are never $L^\alpha$-flat for even $\alpha\ge 4$, which (since sup-norm ultraflatness implies $L^4$-flatness) would answer this problem affirmatively; the claim is not incorporated by the problem page and must be treated as unverified. A Lean formalization of the statement exists in google-deepmind/formal-conjectures. The attacker's tool: rigorous branch-and-bound or exhaustive search over $\pm 1$ coefficient sequences to compute $m(n)=\min_P \max_{\lvert z\rvert=1}\lvert P(z)\rvert$ with certified sup-norm enclosures, charting $m(n)/\sqrt{n}$ as evidence for the optimal $c$; on the proof side, Fourier-analytic lower bounds and independent verification (or refutation) of the claimed $L^4$-flatness route.
References
Investigations · 0
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