{"paper":{"title":"A quantitative inverse theorem for the $U^4$ norm over finite fields","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Luka Mili\\'cevi\\'c, W. T. Gowers","submitted_at":"2017-12-01T09:13:12Z","abstract_excerpt":"A remarkable result of Bergelson, Tao and Ziegler implies that if $c>0$, $k$ is a positive integer, $p\\geq k$ is a prime, $n$ is sufficiently large, and $f:\\mathbb F_p^n\\to\\mathbb C$ is a function with $\\|f\\|_\\infty\\leq 1$ and $\\|f\\|_{U^k}\\geq c$, then there is a polynomial $\\pi$ of degree at most $k-1$ such that $\\mathbb E_xf(x)\\omega^{-\\pi(x)}\\geq c'$, where $\\omega=\\exp(2\\pi i/p)$ and $c'>0$ is a constant that depends on $c,k$ and $p$ only. A version of this result for low-characteristic was also proved by Tao and Ziegler. The proofs of these results do not yield a lower bound for $c'$. Her"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.00241","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":""},"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"}