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We define very weak and very strong regularity for such pairs $(R,G)$ (in the sense of Szemer\\'edi's regularity lemma), and prove that these two properties are equivalent. Moreover, these properties are equivalent to the compactness of the space $(B(H),d_R)/G$. Here $H$ is the completion of $X$ (a Hilbert space), $B(H)$ is the unit ball in $H$, $d_R$ is the metric on $H$ given by $d_R(x,y):=\\sup_{r\\in R}|<r,x-y>|$, and $(B(H),d_R)/G$ is the orbit space "},"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":"1211.3571","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-11-15T10:51:10Z","cross_cats_sorted":[],"title_canon_sha256":"a19d2d9b8ef8e119cd999b7d29b5a05661837f330a3423a4b4889d906c5f5247","abstract_canon_sha256":"b996fb4c7d045607aaa02f9380942cd3858a82084426e7a20ca01a144a72da1a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:40:39.382046Z","signature_b64":"VsJcBoR9K0Fha6jAhaGLg0aXvRClsVNo69f585OJf7cOPeRI/07YbLZHVQ24YEBA5qauoDaEsfk6ij8DhDfZDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bfc6674bf9c0a7eae3c7903beededa50a655a59819588338bc5f2c8b521541ba","last_reissued_at":"2026-05-18T03:40:39.381198Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:40:39.381198Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Weak and strong regularity, compactness, and approximation of polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexander Schrijver","submitted_at":"2012-11-15T10:51:10Z","abstract_excerpt":"Let $X$ be an inner product space, let $G$ be a group of orthogonal transformations of $X$, and let $R$ be a bounded $G$-stable subset of $X$. We define very weak and very strong regularity for such pairs $(R,G)$ (in the sense of Szemer\\'edi's regularity lemma), and prove that these two properties are equivalent. Moreover, these properties are equivalent to the compactness of the space $(B(H),d_R)/G$. 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