{"paper":{"title":"A generalization of Littlewood's $L^\\alpha$ flat theorem, $\\alpha>0$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"el Houcein el Abdalaoui","submitted_at":"2025-09-04T13:37:45Z","abstract_excerpt":"We establish a generalization of Littlewood's criterion on $L^\\alpha$-flatness by proving that there is no $L^\\alpha$-flat polynomials, $\\alpha>0$, within the class of analytic polynomials on the unit circle of the form $ P_n(z)=\\sum_{m=1}^{n}c_m z^m, n \\in {\\mathbb{N}}^*,$ satisfying $$ \\sum_{m=1}^{n}|c_m|^2 \\leq \\frac{K}{n^2} \\sum_{m=1}^{n}m^2 |c_m|^2, $$ where $K$ is an absolutely constant. As a consequence, we confirm the $L^\\alpha$-Littlewood conjecture, and thereby the $L^1$-Newman and $L^\\infty$-Erd\\\"os conjectures. Our approach combines the $L^\\alpha$ Littlewood theorem with the genera"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.04212","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2509.04212/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}