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Abdalaoui 2025 l alpha flatness erdos littlewood s
el Houcein el Abdalaoui, On L^alpha-flatness of Erdős-Littlewood's polynomials. arXiv preprint (2025). arXiv:2504.21499, doi:10.48550/arXiv.2504.21499. The arXiv record (https://arxiv.org/abs/2504.21499, read 2026-10-02) names the Creative Commons Attribution 4.0 license.
The copy read for this card is arXiv version 1 of 30 April 2025, the PDF beside this card. The preprint asserts in Theorem 1 (p. 2) that no sequence of polynomials with plus-or-minus-one coefficients, normalized by 1/sqrt(q), is L^(2p)-flat for any integer p > 1, and in Corollary 1 (p. 2) that none is L^alpha-flat for any alpha >= 4; here a sequence is L^alpha-flat when the moduli converge in L^alpha to 1 (p. 2). The abstract presents this as a partial answer to an old problem of Littlewood, and Corollary 2 (p. 3) as a positive answer to the Erdos-Newman conjecture that no ultraflat sequence of such polynomials exists. The abstract names three inputs: the classical L^p estimate for the Dirichlet kernel (Lemma 1), the interpolation inequalities of Marcinkiewicz and Zygmund (Lemma 3), and the p-concentration theorem of Bonami and Revesz for idempotent polynomials (Lemma 5). If true, Theorem 1 would answer problem 1150 affirmatively, since a sequence of such polynomials with maximum modulus at most (1+o(1)) sqrt(n) would be L^alpha-flat for every finite alpha. In the problem's discussion thread Thomas Bloom reported that the final step of the argument on page 9 does not in fact contradict the paper's own Lemma 5 and that it did not look fixable, and Terence Tao asked that its claims be treated as unconfirmed. Theorem 1.1 of the OpenAI release of 23 September 2026 (card) gives, for every eta > 0 and every large N, sign polynomials of length N with maximum modulus at most (1+eta) sqrt N on the unit circle, hence L^alpha-flat for every finite alpha, so the conclusion of Theorem 1 is false as stated; Appendix A of that release disputes the preprint's argument. The problem's rejected claim page is 2025_04_30_el_abdalaoui. The card records the preprint's statements as claimed; none is verified.
Source: https://arxiv.org/abs/2504.21499.
Bears on. #1150, through the rejected claim page 2025_04_30_el_abdalaoui.