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We use Theorem A to deduce that any affine action $G\\ca^T \\Cal H$ on Hilbert space with weakly mixing linear part admits a sequence of almost fixed points (Theorem B). Specializing to the case that $G$ is a finitely generated group of polynomial growth, we show that convex combinations of averages of the associated 1-cocycle over $n$-balls provide a sequence of"},"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":"1207.5888","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-07-25T05:14:15Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"c908db12ef15048777d98481e28829a88fd6f80b35269e8fb357160454f38227","abstract_canon_sha256":"83c9bc6da3dbb7bda5c0119b5829d5089cb7873dd1036007ab57dbac59f5705a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:49:32.315102Z","signature_b64":"dB7rZBLanAwpcSmr9FUY+K0wLBRKu3k8zsVaC5lGLzIHgZvjd1pBd3/sSVJWlwPjx/Rpm5tHxn5TKuHmrO9vCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5e8fedbe6f53e436a87ad09e060568a69b8816f900d1fac0c5172b5c69a0bb24","last_reissued_at":"2026-05-18T03:49:32.314383Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:49:32.314383Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ergodic theorems for affine actions of amenable groups on Hilbert space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.GR","authors_text":"Ionut Chifan, Thomas Sinclair","submitted_at":"2012-07-25T05:14:15Z","abstract_excerpt":"We prove a new weak mean ergodic theorem (Theorem A) for 1-cocycles associated to weakly mixing representations of amenable groups. 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