{"paper":{"title":"An FKN Theorem for the Binary Grassmann Scheme","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.CC","authors_text":"Anqi Li, Dor Minzer, Yuval Filmus","submitted_at":"2026-08-11T18:12:14Z","abstract_excerpt":"A classical theorem due to Friedgut, Kalai and Naor asserts that if a function $f\\colon \\{0,1\\}^n\\to\\{-1,1\\}$ close to a degree $1$ function, then either $f$ or $-f$ is close to either the all $1$ function, or to $(-1)^{x_i}$ for some $i\\in [n]$. We prove a version of their theorem for the Grassmann scheme over $\\mathbb{F}_2$. More precisely, we prove if a function $f\\colon \\genfrac{[}{]}{0pt}{}{\\mathbb{F}_2^n}{\\ell}\\to\\{0,1\\}$ is close to a degree $1$ function, then either $f$ or $1-f$ must be close to a function of the form $g(L) = \\sum_{x\\in\\mathcal{X}}1_{x\\in L}+\\sum_{W\\in\\mathcal{W}}1_{L\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.11320","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/2608.11320/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"}