{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:QJTDU25EP3FPHM4ZOQYEACYSQA","short_pith_number":"pith:QJTDU25E","canonical_record":{"source":{"id":"1211.2089","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-11-09T10:05:04Z","cross_cats_sorted":["math.FA","math.SP"],"title_canon_sha256":"c216709adf6624fbc50e6421bbe572662e03a364e8ec0b08db5634c4f5aacbc1","abstract_canon_sha256":"468e1e2e0893e3be71a779828cbcd25648589351b129604b294b0d720677a376"},"schema_version":"1.0"},"canonical_sha256":"82663a6ba47ecaf3b3997430400b128013df539763ea69d18fbf5ae410ae1e03","source":{"kind":"arxiv","id":"1211.2089","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.2089","created_at":"2026-05-18T00:32:03Z"},{"alias_kind":"arxiv_version","alias_value":"1211.2089v2","created_at":"2026-05-18T00:32:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.2089","created_at":"2026-05-18T00:32:03Z"},{"alias_kind":"pith_short_12","alias_value":"QJTDU25EP3FP","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"QJTDU25EP3FPHM4Z","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"QJTDU25E","created_at":"2026-05-18T12:27:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:QJTDU25EP3FPHM4ZOQYEACYSQA","target":"record","payload":{"canonical_record":{"source":{"id":"1211.2089","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-11-09T10:05:04Z","cross_cats_sorted":["math.FA","math.SP"],"title_canon_sha256":"c216709adf6624fbc50e6421bbe572662e03a364e8ec0b08db5634c4f5aacbc1","abstract_canon_sha256":"468e1e2e0893e3be71a779828cbcd25648589351b129604b294b0d720677a376"},"schema_version":"1.0"},"canonical_sha256":"82663a6ba47ecaf3b3997430400b128013df539763ea69d18fbf5ae410ae1e03","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:32:03.427853Z","signature_b64":"igQiftMAwuEFDwX1iF6ETowaYmcIyqebLVOa0yqsmceYoiCmnmwZpKpFvP3Q7T4swEfX35oN5OKwPUdanaQLBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"82663a6ba47ecaf3b3997430400b128013df539763ea69d18fbf5ae410ae1e03","last_reissued_at":"2026-05-18T00:32:03.427370Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:32:03.427370Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1211.2089","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:32:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IjZz7Xm9II9Ax66nnOH+wgpWAZQ7cXr9cVSOywk1dSK8HlZEftqvPgu11exMN3xtAXpI48Birtj+lzNkIYS9Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T15:35:13.595552Z"},"content_sha256":"db04c163e992b5c5e0c2490a97822a7577390624bd9c5190241023cc189215ed","schema_version":"1.0","event_id":"sha256:db04c163e992b5c5e0c2490a97822a7577390624bd9c5190241023cc189215ed"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:QJTDU25EP3FPHM4ZOQYEACYSQA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Almost-additive ergodic theorems for amenable groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.SP"],"primary_cat":"math.DS","authors_text":"Felix Pogorzelski","submitted_at":"2012-11-09T10:05:04Z","abstract_excerpt":"In this paper we prove a general convergence theorem for almost-additive set functions on unimodular, amenable groups. These mappings take their values in some Banach space. By extending the theory of epsilon-quasi tiling techniques, we set the ground for far-reaching applications in the theory of group dynamics. In particular, we verify the almost-everywhere convergence of abstract approximable bounded, additive processes, as well as a Banach space approximation result for the spectral distribution function (integrated density of states) for random operators on discrete structures in a metric"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.2089","kind":"arxiv","version":2},"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":"2026-05-18T00:32:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8vN9Z55H4qd1uQarF/oXTOlV3VAE1ycR0ygN6smMbWoOmlvuky72DULxF73upJKThQyS4MU3VRHK1KMzSwtHDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T15:35:13.596176Z"},"content_sha256":"5721038fe1db84bab36cbec8647b90b4193e3020767933923ed7ceed02eb6673","schema_version":"1.0","event_id":"sha256:5721038fe1db84bab36cbec8647b90b4193e3020767933923ed7ceed02eb6673"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QJTDU25EP3FPHM4ZOQYEACYSQA/bundle.json","state_url":"https://pith.science/pith/QJTDU25EP3FPHM4ZOQYEACYSQA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QJTDU25EP3FPHM4ZOQYEACYSQA/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":"2026-05-27T15:35:13Z","links":{"resolver":"https://pith.science/pith/QJTDU25EP3FPHM4ZOQYEACYSQA","bundle":"https://pith.science/pith/QJTDU25EP3FPHM4ZOQYEACYSQA/bundle.json","state":"https://pith.science/pith/QJTDU25EP3FPHM4ZOQYEACYSQA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QJTDU25EP3FPHM4ZOQYEACYSQA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:QJTDU25EP3FPHM4ZOQYEACYSQA","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":"468e1e2e0893e3be71a779828cbcd25648589351b129604b294b0d720677a376","cross_cats_sorted":["math.FA","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-11-09T10:05:04Z","title_canon_sha256":"c216709adf6624fbc50e6421bbe572662e03a364e8ec0b08db5634c4f5aacbc1"},"schema_version":"1.0","source":{"id":"1211.2089","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.2089","created_at":"2026-05-18T00:32:03Z"},{"alias_kind":"arxiv_version","alias_value":"1211.2089v2","created_at":"2026-05-18T00:32:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.2089","created_at":"2026-05-18T00:32:03Z"},{"alias_kind":"pith_short_12","alias_value":"QJTDU25EP3FP","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"QJTDU25EP3FPHM4Z","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"QJTDU25E","created_at":"2026-05-18T12:27:18Z"}],"graph_snapshots":[{"event_id":"sha256:5721038fe1db84bab36cbec8647b90b4193e3020767933923ed7ceed02eb6673","target":"graph","created_at":"2026-05-18T00:32:03Z","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":"In this paper we prove a general convergence theorem for almost-additive set functions on unimodular, amenable groups. These mappings take their values in some Banach space. By extending the theory of epsilon-quasi tiling techniques, we set the ground for far-reaching applications in the theory of group dynamics. In particular, we verify the almost-everywhere convergence of abstract approximable bounded, additive processes, as well as a Banach space approximation result for the spectral distribution function (integrated density of states) for random operators on discrete structures in a metric","authors_text":"Felix Pogorzelski","cross_cats":["math.FA","math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-11-09T10:05:04Z","title":"Almost-additive ergodic theorems for amenable groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.2089","kind":"arxiv","version":2},"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:db04c163e992b5c5e0c2490a97822a7577390624bd9c5190241023cc189215ed","target":"record","created_at":"2026-05-18T00:32:03Z","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":"468e1e2e0893e3be71a779828cbcd25648589351b129604b294b0d720677a376","cross_cats_sorted":["math.FA","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2012-11-09T10:05:04Z","title_canon_sha256":"c216709adf6624fbc50e6421bbe572662e03a364e8ec0b08db5634c4f5aacbc1"},"schema_version":"1.0","source":{"id":"1211.2089","kind":"arxiv","version":2}},"canonical_sha256":"82663a6ba47ecaf3b3997430400b128013df539763ea69d18fbf5ae410ae1e03","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"82663a6ba47ecaf3b3997430400b128013df539763ea69d18fbf5ae410ae1e03","first_computed_at":"2026-05-18T00:32:03.427370Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:32:03.427370Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"igQiftMAwuEFDwX1iF6ETowaYmcIyqebLVOa0yqsmceYoiCmnmwZpKpFvP3Q7T4swEfX35oN5OKwPUdanaQLBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:32:03.427853Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.2089","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:db04c163e992b5c5e0c2490a97822a7577390624bd9c5190241023cc189215ed","sha256:5721038fe1db84bab36cbec8647b90b4193e3020767933923ed7ceed02eb6673"],"state_sha256":"f9ef71f5f533cea09dc59b6ba31ecf34f84b3d8b6260ee067b0a4e90b120fbba"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"W9IXrxyjvn4lLdwNK1+yQ1daol8je43nl37h31gmf8CW5hOXG3L/fK5ok9F7gneM4oayrmNaSTKe9JAM+F/RCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T15:35:13.600425Z","bundle_sha256":"0df42f2e3396994c8a2431d9d7acfb3751146f2a51ae73e8aebf3df395f9cf4e"}}