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This result shows that every degree-d PTF can be decomposed into a constant number of subfunctions such that almost all of the subfunctions are close to being regular PTFs. Here a \"regular PTF is a PTF sign(p(x)) where the influence of each variable on the polynomial p(x) is a small fraction of the total influence of p.\n  As an application of this regularity lemma, we prove that for any constants d \\geq 1, \\eps \\geq 0, every degree-d PTF over n variables has can be approximated to ac"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0909.4727","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2009-09-25T15:54:33Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"08bb94054fad0edf6a4065faba42da24933c72cfe27a22f1402d0504815a374d","abstract_canon_sha256":"1d830fa3ee422a58c7bed4cb7e4ab372c030a888db291fffb49461b23d0f67b0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:24:47.066177Z","signature_b64":"+A6rlgbwa25j8bgVL2LAajIbu0YJ7elFGOjTXXjY55XfD+Ph5DRVtb/69B2vga5QBVtIZ6TV0snKc6QQPWn1Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f62c1779a8033edc0c57568fc3418982674a95013528c90c45950a83a7ed2790","last_reissued_at":"2026-05-18T02:24:47.065704Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:24:47.065704Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A regularity lemma, and low-weight approximators, for low-degree polynomial threshold functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"cs.CC","authors_text":"Andrew Wan, Ilias Diakonikolas, Li-Yang Tan, Rocco A. 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