{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:3YGJASXWFRUE2JYWQUM7ZZLNND","short_pith_number":"pith:3YGJASXW","canonical_record":{"source":{"id":"1706.07954","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-06-24T13:34:06Z","cross_cats_sorted":["math.FA","math.GN","math.NT","math.PR"],"title_canon_sha256":"ed16a54b1ccadbc5b0ceedced7951434dbfc36300054108bcbc652a134c5c57a","abstract_canon_sha256":"1813c94d43dd4f20b75f51def97e105b4a4fda8020a7c03efd815c40d0a9403d"},"schema_version":"1.0"},"canonical_sha256":"de0c904af62c684d27168519fce56d68d035c6bfb5786794844f3c8e3430fdd5","source":{"kind":"arxiv","id":"1706.07954","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.07954","created_at":"2026-05-18T00:24:35Z"},{"alias_kind":"arxiv_version","alias_value":"1706.07954v4","created_at":"2026-05-18T00:24:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.07954","created_at":"2026-05-18T00:24:35Z"},{"alias_kind":"pith_short_12","alias_value":"3YGJASXWFRUE","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"3YGJASXWFRUE2JYW","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"3YGJASXW","created_at":"2026-05-18T12:30:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:3YGJASXWFRUE2JYWQUM7ZZLNND","target":"record","payload":{"canonical_record":{"source":{"id":"1706.07954","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-06-24T13:34:06Z","cross_cats_sorted":["math.FA","math.GN","math.NT","math.PR"],"title_canon_sha256":"ed16a54b1ccadbc5b0ceedced7951434dbfc36300054108bcbc652a134c5c57a","abstract_canon_sha256":"1813c94d43dd4f20b75f51def97e105b4a4fda8020a7c03efd815c40d0a9403d"},"schema_version":"1.0"},"canonical_sha256":"de0c904af62c684d27168519fce56d68d035c6bfb5786794844f3c8e3430fdd5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:24:35.320306Z","signature_b64":"K5MjpCkBsOIIgKbrYK0OENSLx8OMkFPJcSDMFmbj067SrgYxHz9nnPCZDYmIEdsuAcyUvuJQHhEdOBBi06JsDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"de0c904af62c684d27168519fce56d68d035c6bfb5786794844f3c8e3430fdd5","last_reissued_at":"2026-05-18T00:24:35.319872Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:24:35.319872Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1706.07954","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:24:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3NtL8+SWKhzlu2+iHD8upJSSDOjLmPbygrlqaC4INR31KpTQhPHl1vk+9RRt4VW1gyB0KNvQ8DXMpsrEiGQCDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T11:10:00.434534Z"},"content_sha256":"9a7d815384efb8b3fa71dbc2bc9ea46abebd344071cd85d83ee69a142192ed65","schema_version":"1.0","event_id":"sha256:9a7d815384efb8b3fa71dbc2bc9ea46abebd344071cd85d83ee69a142192ed65"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:3YGJASXWFRUE2JYWQUM7ZZLNND","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Thinnable Ideals and Invariance of Cluster Points","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.GN","math.NT","math.PR"],"primary_cat":"math.CA","authors_text":"Paolo Leonetti","submitted_at":"2017-06-24T13:34:06Z","abstract_excerpt":"We define a class of so-called thinnable ideals $\\mathcal{I}$ on the positive integers which includes several well-known examples, e.g., the collection of sets with zero asymptotic density, sets with zero logarithmic density, and several summable ideals. Given a sequence $(x_n)$ taking values in a separable metric space and a thinnable ideal $\\mathcal{I}$, it is shown that the set of $\\mathcal{I}$-cluster points of $(x_n)$ is equal to the set of $\\mathcal{I}$-cluster points of almost all its subsequences, in the sense of Lebesgue measure. Lastly, we obtain a characterization of ideal convergen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.07954","kind":"arxiv","version":4},"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":"2026-05-18T00:24:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"l/5oXQHfmn4hDU13DA59WWQjjMvKJNFBw/P7FywkSHKptokJlkCX+5zx45npRPKfQ6pZuutN3Kv71qSqBgdwCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T11:10:00.434884Z"},"content_sha256":"e9682ea42d14dac90c00e91383f9438b2e3759143bce01281dbec4c49478726a","schema_version":"1.0","event_id":"sha256:e9682ea42d14dac90c00e91383f9438b2e3759143bce01281dbec4c49478726a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3YGJASXWFRUE2JYWQUM7ZZLNND/bundle.json","state_url":"https://pith.science/pith/3YGJASXWFRUE2JYWQUM7ZZLNND/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3YGJASXWFRUE2JYWQUM7ZZLNND/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":"2026-07-03T11:10:00Z","links":{"resolver":"https://pith.science/pith/3YGJASXWFRUE2JYWQUM7ZZLNND","bundle":"https://pith.science/pith/3YGJASXWFRUE2JYWQUM7ZZLNND/bundle.json","state":"https://pith.science/pith/3YGJASXWFRUE2JYWQUM7ZZLNND/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3YGJASXWFRUE2JYWQUM7ZZLNND/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:3YGJASXWFRUE2JYWQUM7ZZLNND","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":"1813c94d43dd4f20b75f51def97e105b4a4fda8020a7c03efd815c40d0a9403d","cross_cats_sorted":["math.FA","math.GN","math.NT","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-06-24T13:34:06Z","title_canon_sha256":"ed16a54b1ccadbc5b0ceedced7951434dbfc36300054108bcbc652a134c5c57a"},"schema_version":"1.0","source":{"id":"1706.07954","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.07954","created_at":"2026-05-18T00:24:35Z"},{"alias_kind":"arxiv_version","alias_value":"1706.07954v4","created_at":"2026-05-18T00:24:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.07954","created_at":"2026-05-18T00:24:35Z"},{"alias_kind":"pith_short_12","alias_value":"3YGJASXWFRUE","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"3YGJASXWFRUE2JYW","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"3YGJASXW","created_at":"2026-05-18T12:30:58Z"}],"graph_snapshots":[{"event_id":"sha256:e9682ea42d14dac90c00e91383f9438b2e3759143bce01281dbec4c49478726a","target":"graph","created_at":"2026-05-18T00:24:35Z","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 define a class of so-called thinnable ideals $\\mathcal{I}$ on the positive integers which includes several well-known examples, e.g., the collection of sets with zero asymptotic density, sets with zero logarithmic density, and several summable ideals. Given a sequence $(x_n)$ taking values in a separable metric space and a thinnable ideal $\\mathcal{I}$, it is shown that the set of $\\mathcal{I}$-cluster points of $(x_n)$ is equal to the set of $\\mathcal{I}$-cluster points of almost all its subsequences, in the sense of Lebesgue measure. Lastly, we obtain a characterization of ideal convergen","authors_text":"Paolo Leonetti","cross_cats":["math.FA","math.GN","math.NT","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-06-24T13:34:06Z","title":"Thinnable Ideals and Invariance of Cluster Points"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.07954","kind":"arxiv","version":4},"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":"sha256:9a7d815384efb8b3fa71dbc2bc9ea46abebd344071cd85d83ee69a142192ed65","target":"record","created_at":"2026-05-18T00:24:35Z","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":"1813c94d43dd4f20b75f51def97e105b4a4fda8020a7c03efd815c40d0a9403d","cross_cats_sorted":["math.FA","math.GN","math.NT","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2017-06-24T13:34:06Z","title_canon_sha256":"ed16a54b1ccadbc5b0ceedced7951434dbfc36300054108bcbc652a134c5c57a"},"schema_version":"1.0","source":{"id":"1706.07954","kind":"arxiv","version":4}},"canonical_sha256":"de0c904af62c684d27168519fce56d68d035c6bfb5786794844f3c8e3430fdd5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"de0c904af62c684d27168519fce56d68d035c6bfb5786794844f3c8e3430fdd5","first_computed_at":"2026-05-18T00:24:35.319872Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:24:35.319872Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"K5MjpCkBsOIIgKbrYK0OENSLx8OMkFPJcSDMFmbj067SrgYxHz9nnPCZDYmIEdsuAcyUvuJQHhEdOBBi06JsDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:24:35.320306Z","signed_message":"canonical_sha256_bytes"},"source_id":"1706.07954","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9a7d815384efb8b3fa71dbc2bc9ea46abebd344071cd85d83ee69a142192ed65","sha256:e9682ea42d14dac90c00e91383f9438b2e3759143bce01281dbec4c49478726a"],"state_sha256":"46f4025728a0530b75b5b1be336772cdcf017604e568a77c63d218b4acc9d64e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X8Go2I4FlN1kFfo0GsVYGiFWPD0CyImPSzpX+NPEaktQkdFEOCY+W2S27OPvfR0htI9CVDGgHSmDUrssLsWDDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-03T11:10:00.436894Z","bundle_sha256":"ac70c9feffa9308653f441108f7e3c7d4d17397333a716fa4a7d744cd4094093"}}