{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:A5HMYKQAMEYLWARIMB2TN44PJH","short_pith_number":"pith:A5HMYKQA","canonical_record":{"source":{"id":"1408.6692","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-08-28T12:11:08Z","cross_cats_sorted":["math.GR","math.PR"],"title_canon_sha256":"9cd13e0e54c5d6341563a046ef0262a0a9483de3b102dedf14fa34d504ab1c71","abstract_canon_sha256":"ed65e70220b2e2584dc1820268766945e595091d86809d0f56f9b3ab4b9b1f47"},"schema_version":"1.0"},"canonical_sha256":"074ecc2a006130bb0228607536f38f49dbd3ad38abd30baf111e158c655e3bbf","source":{"kind":"arxiv","id":"1408.6692","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.6692","created_at":"2026-05-18T02:44:01Z"},{"alias_kind":"arxiv_version","alias_value":"1408.6692v1","created_at":"2026-05-18T02:44:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.6692","created_at":"2026-05-18T02:44:01Z"},{"alias_kind":"pith_short_12","alias_value":"A5HMYKQAMEYL","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_16","alias_value":"A5HMYKQAMEYLWARI","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_8","alias_value":"A5HMYKQA","created_at":"2026-05-18T12:28:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:A5HMYKQAMEYLWARIMB2TN44PJH","target":"record","payload":{"canonical_record":{"source":{"id":"1408.6692","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-08-28T12:11:08Z","cross_cats_sorted":["math.GR","math.PR"],"title_canon_sha256":"9cd13e0e54c5d6341563a046ef0262a0a9483de3b102dedf14fa34d504ab1c71","abstract_canon_sha256":"ed65e70220b2e2584dc1820268766945e595091d86809d0f56f9b3ab4b9b1f47"},"schema_version":"1.0"},"canonical_sha256":"074ecc2a006130bb0228607536f38f49dbd3ad38abd30baf111e158c655e3bbf","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:44:01.040905Z","signature_b64":"pubutLD+oQhL2+6xDQ17W5/zOHKM9j6z+vDori8WRTGZTBqAW6zMrdy39X2DTgDcxNenxje36Bq09MNA3ilACw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"074ecc2a006130bb0228607536f38f49dbd3ad38abd30baf111e158c655e3bbf","last_reissued_at":"2026-05-18T02:44:01.040422Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:44:01.040422Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1408.6692","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:44:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y6fs88TqWDYK9hCYfF5d2wJlHAz6yWzwfksvcUgMRoDfUtbmx3a4LqQkldMSRldyjSILQ+ne3HTCXglr5AeFAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T01:26:19.013699Z"},"content_sha256":"ebc7d26271a64559920d24293f0fc9759e8e90d668a5fd58d8d4006bfab98b13","schema_version":"1.0","event_id":"sha256:ebc7d26271a64559920d24293f0fc9759e8e90d668a5fd58d8d4006bfab98b13"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:A5HMYKQAMEYLWARIMB2TN44PJH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Ergodic Theorems for coset spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR","math.PR"],"primary_cat":"math.DS","authors_text":"Alexander Fish, Michael Bj\\\"orklund","submitted_at":"2014-08-28T12:11:08Z","abstract_excerpt":"We study in this paper the validity of the mean ergodic theorem along \\emph{left} F\\o lner sequences in a countable amenable group $G$. Although the \\emph{weak} ergodic theorem always holds along \\emph{any} left F\\o lner sequence in $G$, we provide examples where the \\emph{mean} ergodic theorem fails in quite dramatic ways. On the other hand, if $G$ does not admit any ICC quotients, e.g. if $G$ is virtually nilpotent, then we prove that the mean ergodic theorem does indeed hold along \\emph{any} left F\\o lner sequence.\n  In the case when a unitary representation of a countable amenable group is"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.6692","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:44:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lgL0/kRwEE0rtx+I2Di9CaBLQTOMMgw76TIv8VmaLVPiYk8Ks4WdznOBJcSYSDxuB2kj6CzZ70rl6f6Ou80UCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T01:26:19.014395Z"},"content_sha256":"ebdc8b50f1371e742cc4b9a273bea1d2cd636b805ebca0bb4c3c0bbfa64ae103","schema_version":"1.0","event_id":"sha256:ebdc8b50f1371e742cc4b9a273bea1d2cd636b805ebca0bb4c3c0bbfa64ae103"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/A5HMYKQAMEYLWARIMB2TN44PJH/bundle.json","state_url":"https://pith.science/pith/A5HMYKQAMEYLWARIMB2TN44PJH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/A5HMYKQAMEYLWARIMB2TN44PJH/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-10T01:26:19Z","links":{"resolver":"https://pith.science/pith/A5HMYKQAMEYLWARIMB2TN44PJH","bundle":"https://pith.science/pith/A5HMYKQAMEYLWARIMB2TN44PJH/bundle.json","state":"https://pith.science/pith/A5HMYKQAMEYLWARIMB2TN44PJH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/A5HMYKQAMEYLWARIMB2TN44PJH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:A5HMYKQAMEYLWARIMB2TN44PJH","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"ed65e70220b2e2584dc1820268766945e595091d86809d0f56f9b3ab4b9b1f47","cross_cats_sorted":["math.GR","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-08-28T12:11:08Z","title_canon_sha256":"9cd13e0e54c5d6341563a046ef0262a0a9483de3b102dedf14fa34d504ab1c71"},"schema_version":"1.0","source":{"id":"1408.6692","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1408.6692","created_at":"2026-05-18T02:44:01Z"},{"alias_kind":"arxiv_version","alias_value":"1408.6692v1","created_at":"2026-05-18T02:44:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1408.6692","created_at":"2026-05-18T02:44:01Z"},{"alias_kind":"pith_short_12","alias_value":"A5HMYKQAMEYL","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_16","alias_value":"A5HMYKQAMEYLWARI","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_8","alias_value":"A5HMYKQA","created_at":"2026-05-18T12:28:19Z"}],"graph_snapshots":[{"event_id":"sha256:ebdc8b50f1371e742cc4b9a273bea1d2cd636b805ebca0bb4c3c0bbfa64ae103","target":"graph","created_at":"2026-05-18T02:44:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We study in this paper the validity of the mean ergodic theorem along \\emph{left} F\\o lner sequences in a countable amenable group $G$. Although the \\emph{weak} ergodic theorem always holds along \\emph{any} left F\\o lner sequence in $G$, we provide examples where the \\emph{mean} ergodic theorem fails in quite dramatic ways. On the other hand, if $G$ does not admit any ICC quotients, e.g. if $G$ is virtually nilpotent, then we prove that the mean ergodic theorem does indeed hold along \\emph{any} left F\\o lner sequence.\n  In the case when a unitary representation of a countable amenable group is","authors_text":"Alexander Fish, Michael Bj\\\"orklund","cross_cats":["math.GR","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-08-28T12:11:08Z","title":"Ergodic Theorems for coset spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.6692","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:ebc7d26271a64559920d24293f0fc9759e8e90d668a5fd58d8d4006bfab98b13","target":"record","created_at":"2026-05-18T02:44:01Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"ed65e70220b2e2584dc1820268766945e595091d86809d0f56f9b3ab4b9b1f47","cross_cats_sorted":["math.GR","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2014-08-28T12:11:08Z","title_canon_sha256":"9cd13e0e54c5d6341563a046ef0262a0a9483de3b102dedf14fa34d504ab1c71"},"schema_version":"1.0","source":{"id":"1408.6692","kind":"arxiv","version":1}},"canonical_sha256":"074ecc2a006130bb0228607536f38f49dbd3ad38abd30baf111e158c655e3bbf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"074ecc2a006130bb0228607536f38f49dbd3ad38abd30baf111e158c655e3bbf","first_computed_at":"2026-05-18T02:44:01.040422Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:44:01.040422Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"pubutLD+oQhL2+6xDQ17W5/zOHKM9j6z+vDori8WRTGZTBqAW6zMrdy39X2DTgDcxNenxje36Bq09MNA3ilACw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:44:01.040905Z","signed_message":"canonical_sha256_bytes"},"source_id":"1408.6692","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ebc7d26271a64559920d24293f0fc9759e8e90d668a5fd58d8d4006bfab98b13","sha256:ebdc8b50f1371e742cc4b9a273bea1d2cd636b805ebca0bb4c3c0bbfa64ae103"],"state_sha256":"b49c7c850b346d4e670b506e25b72e79a48b3f42eb61054b37269ca67c272dcd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"b46VdpkXL9lLvGRe18kWzHcURNLR6ICYa/3dmEnPIDWFBIXwD4E+qfZYIQaVj+0C0RmtfddHhf6gED5hmQ7fCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T01:26:19.019956Z","bundle_sha256":"f92d0f66ad9e66338c4f3861f4df98bf996d3c1b8e9be3d306f151d126ceb23f"}}