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Claim. The preprint el Houcein el Abdalaoui, On $L^\alpha$-flatness of Erdős–Littlewood's polynomials, arXiv:2504.21499 (one version, posted 2025-04-30), states that the polynomials with coefficients $\pm1$ are not $L^\alpha$-flat for any even integer $\alpha>2$, hence for every $\alpha\ge4$: for such $\alpha$ the normalized norms $\lVert P\rVert_\alpha/\sqrt{n+1}$ of the $\pm1$ polynomials of degree $n$ stay bounded away from $1$ as $n\to\infty$. The author presents this as a positive answer to the Erdős–Newman conjecture that no ultraflat sequence of $\pm1$ polynomials exists. Since $\max_{\lvert z\rvert=1}\lvert P(z)\rvert\ge\lVert P\rVert_4$, the claim would give a constant $c>0$ with $\max_{\lvert z\rvert=1}\lvert P(z)\rvert>(1+c)\sqrt n$ for every $\pm1$ polynomial of every large degree $n$, which is Problem 1150 answered yes. The stated inputs are the $L^p$ bounds for the Dirichlet kernel, the Marcinkiewicz–Zygmund interpolation inequalities and the $p$-concentration theorem of Bonami and Révész; the preprint is paged at its card. A reader cited it in the problem's discussion thread on 2026-02-04 as answering the problem affirmatively. The author first claimed the affirmative answer in arXiv:1609.03435 (2016), whose Theorem 3.3 is the case $\alpha=4$ of the claim above. The author claimed it again in arXiv:2509.04212 (September 2025) for every $\alpha>0$. Appendix A of the release treats each separately, and each has its own rejected page (2016, September 2025). The author's other earlier preprints concern restricted classes or Newman polynomials and post no answer to this problem.

Depends on. No page of this wiki for the claim itself; the rejection below rests on the accepted OpenAI 2026 page.

Rejection. The claim is rejected on three grounds. In the thread, on 2026-02-04, the site's curator, Thomas F. Bloom, replied that the final step of the proof on p. 9 does not contradict the preprint's Lemma 5, which only gives one function with concentration, and that the argument did not look fixable; Tao replied the same day that the preprint relies on the author's earlier unpublished preprints and that its claims should be treated as unconfirmed until publication or independent verification. The author answered in the thread on 2026-08-16 with references and clarifications and asked for further feedback; no corrected version is posted. Second, the accepted Theorem 1.1 of the OpenAI release contradicts the conclusion: for every $\eta>0$ and every large $N$ it gives signs with $\max_{\lvert z\rvert=1}\lvert P(z)\rvert\le(1+\eta)\sqrt N$, and since $\lVert P\rVert_2=\sqrt N$ by Parseval, Hölder's inequality $\lVert P\rVert_\alpha\le\lVert P\rVert_\infty^{1-2/\alpha}\lVert P\rVert_2^{2/\alpha}$ puts $\lVert P\rVert_\alpha/\sqrt N$ between $1$ and $(1+\eta)^{1-2/\alpha}$ for every finite $\alpha\ge2$, so these polynomials are $L^\alpha$-flat for every even $\alpha>2$. Third, Appendix A of the release manuscript (card) locates the failing inference in the preprint's argument. Not reviewed, not refereed: the preprint has no journal version, and the site's label is OPEN (page last edited 23 January 2026, accessed 2026-10-06).