{"paper":{"title":"Sharp estimates for Gowers norms on discrete cubes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.CA","math.IT"],"primary_cat":"math.CO","authors_text":"Adrian Beker, Ton\\'ci Crmari\\'c, Vjekoslav Kova\\v{c}","submitted_at":"2024-09-19T09:00:20Z","abstract_excerpt":"We study optimal dimensionless inequalities $$ \\|f\\|_{U^k} \\leq \\|f\\|_{\\ell^{p_{k,n}}} $$ that hold for all functions $f\\colon\\mathbb{Z}^d\\to\\mathbb{C}$ supported in $\\{0,1,\\ldots,n-1\\}^d$ and estimates $$ \\|1_A\\|_{U^k}^{2^k}\\leq |A|^{t_{k,n}} $$ that hold for all subsets $A$ of the same discrete cubes. A general theory, analogous to the work of de Dios Pont, Greenfeld, Ivanisvili, and Madrid, is developed to show that the critical exponents are related by $p_{k,n} t_{k,n} = 2^k$. This is used to prove the three main results of the paper: an explicit formula for $t_{k,2}$, which generalizes a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.12579","kind":"arxiv","version":2},"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/2409.12579/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"}