On $L^α$-flatness of Erdős-Littlewood's polynomials
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866916714773479424 |
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| author | Abdalaoui, el Houcein el |
| author_facet | Abdalaoui, el Houcein el |
| contents | It is shown that Erdös--Littlewood's polynomials are not $L^α$-flat when $α> 2$ is an even integer (and hence for any $α\geq 4$). This provides a partial solution to an old problem posed by Littlewood. Consequently, we obtain a positive answer to the analogous Erdös--Newman conjecture for polynomials with coefficients $\pm 1$; that is, there is no ultraflat sequence of polynomials from the class of Erdös--Littlewood polynomials.
Our proof is short and simple. It relies on the classical lemma for $L^p$ norms of the Dirichlet kernel, the Marcinkiewicz--Zygmund interpolation inequalities, and the $p$-concentration theorem due to A. Bonami and S. Révész. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21499 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On $L^α$-flatness of Erdős-Littlewood's polynomials Abdalaoui, el Houcein el Classical Analysis and ODEs Dynamical Systems 42A05, 42A55 (Primary), 37A05, 37A30 (Secondary) It is shown that Erdös--Littlewood's polynomials are not $L^α$-flat when $α> 2$ is an even integer (and hence for any $α\geq 4$). This provides a partial solution to an old problem posed by Littlewood. Consequently, we obtain a positive answer to the analogous Erdös--Newman conjecture for polynomials with coefficients $\pm 1$; that is, there is no ultraflat sequence of polynomials from the class of Erdös--Littlewood polynomials. Our proof is short and simple. It relies on the classical lemma for $L^p$ norms of the Dirichlet kernel, the Marcinkiewicz--Zygmund interpolation inequalities, and the $p$-concentration theorem due to A. Bonami and S. Révész. |
| title | On $L^α$-flatness of Erdős-Littlewood's polynomials |
| topic | Classical Analysis and ODEs Dynamical Systems 42A05, 42A55 (Primary), 37A05, 37A30 (Secondary) |
| url | https://arxiv.org/abs/2504.21499 |