{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:3N7HZK6SBOSV6N7JLFIJ76M65V","short_pith_number":"pith:3N7HZK6S","canonical_record":{"source":{"id":"2006.02066","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2020-06-03T06:41:23Z","cross_cats_sorted":[],"title_canon_sha256":"649fe671b7aad99d67efb9184e7e55a786364c60242106c659a92edab95b6950","abstract_canon_sha256":"daed14782ebb7f3bd95e64d69a8eed3f3709642ef0769d108dab7c1e8cbd81a7"},"schema_version":"1.0"},"canonical_sha256":"db7e7cabd20ba55f37e959509ff99eed6002107f90e04187f24de1779a2fc0bd","source":{"kind":"arxiv","id":"2006.02066","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2006.02066","created_at":"2026-07-05T01:51:19Z"},{"alias_kind":"arxiv_version","alias_value":"2006.02066v2","created_at":"2026-07-05T01:51:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2006.02066","created_at":"2026-07-05T01:51:19Z"},{"alias_kind":"pith_short_12","alias_value":"3N7HZK6SBOSV","created_at":"2026-07-05T01:51:19Z"},{"alias_kind":"pith_short_16","alias_value":"3N7HZK6SBOSV6N7J","created_at":"2026-07-05T01:51:19Z"},{"alias_kind":"pith_short_8","alias_value":"3N7HZK6S","created_at":"2026-07-05T01:51:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:3N7HZK6SBOSV6N7JLFIJ76M65V","target":"record","payload":{"canonical_record":{"source":{"id":"2006.02066","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2020-06-03T06:41:23Z","cross_cats_sorted":[],"title_canon_sha256":"649fe671b7aad99d67efb9184e7e55a786364c60242106c659a92edab95b6950","abstract_canon_sha256":"daed14782ebb7f3bd95e64d69a8eed3f3709642ef0769d108dab7c1e8cbd81a7"},"schema_version":"1.0"},"canonical_sha256":"db7e7cabd20ba55f37e959509ff99eed6002107f90e04187f24de1779a2fc0bd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:51:19.251400Z","signature_b64":"mV/Czs0VhuOQY4NDXZJDTK0aRjsG7YU3tsP90sth+vC6/5oySt4SVgYUkYhWW5o7fGoQ/KC1/E0kW9fGrCN4Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"db7e7cabd20ba55f37e959509ff99eed6002107f90e04187f24de1779a2fc0bd","last_reissued_at":"2026-07-05T01:51:19.250977Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:51:19.250977Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2006.02066","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":"2026-07-05T01:51:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O71wh1cR9Z5Nl1InKceCUL1zX9WiGYrYdXEZjNCmrpP/dr42U6atl04ba/SsHEhSM74FmxhRK7lKg8uYygokCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T16:01:02.918815Z"},"content_sha256":"d82053b5c89f97c45ee029b542435fb464f773ff7f88ab19888bedf81e7dae40","schema_version":"1.0","event_id":"sha256:d82053b5c89f97c45ee029b542435fb464f773ff7f88ab19888bedf81e7dae40"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:3N7HZK6SBOSV6N7JLFIJ76M65V","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the limits of real-valued functions in sets involving $ \\psi $-density, and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"J. Heittokangas, J. Wang, M. A. Zemirni, Z. Latreuch","submitted_at":"2020-06-03T06:41:23Z","abstract_excerpt":"We prove new results on upper and lower limits of real-valued functions by means of $\\psi$-densities introduced by P. D. Barry in 1962. This allows us to improve several existing results on the growth of non-decreasing and unbounded real-valued functions in sets of positive density. The $\\psi$-densities are also used to introduce a new concept of a limit for real-valued functions. The results in this paper are of interest in real analysis as well as in the theory of meromorphic functions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.02066","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2006.02066/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T01:51:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WOI4mzB5NVqPrMg3X/af+HH0gkAXrh3mvCKO/SkDOJkXe7zDqPB2+yE5sOIUqZfBk4oJbgcNhWXzoCfXVmSNAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-07T16:01:02.919199Z"},"content_sha256":"3ab6f1f73960f48615a4018019c68a5be8d79bfbf79ad006c3180a87c0d46445","schema_version":"1.0","event_id":"sha256:3ab6f1f73960f48615a4018019c68a5be8d79bfbf79ad006c3180a87c0d46445"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3N7HZK6SBOSV6N7JLFIJ76M65V/bundle.json","state_url":"https://pith.science/pith/3N7HZK6SBOSV6N7JLFIJ76M65V/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3N7HZK6SBOSV6N7JLFIJ76M65V/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":"2026-07-07T16:01:02Z","links":{"resolver":"https://pith.science/pith/3N7HZK6SBOSV6N7JLFIJ76M65V","bundle":"https://pith.science/pith/3N7HZK6SBOSV6N7JLFIJ76M65V/bundle.json","state":"https://pith.science/pith/3N7HZK6SBOSV6N7JLFIJ76M65V/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3N7HZK6SBOSV6N7JLFIJ76M65V/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:3N7HZK6SBOSV6N7JLFIJ76M65V","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":"daed14782ebb7f3bd95e64d69a8eed3f3709642ef0769d108dab7c1e8cbd81a7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2020-06-03T06:41:23Z","title_canon_sha256":"649fe671b7aad99d67efb9184e7e55a786364c60242106c659a92edab95b6950"},"schema_version":"1.0","source":{"id":"2006.02066","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2006.02066","created_at":"2026-07-05T01:51:19Z"},{"alias_kind":"arxiv_version","alias_value":"2006.02066v2","created_at":"2026-07-05T01:51:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2006.02066","created_at":"2026-07-05T01:51:19Z"},{"alias_kind":"pith_short_12","alias_value":"3N7HZK6SBOSV","created_at":"2026-07-05T01:51:19Z"},{"alias_kind":"pith_short_16","alias_value":"3N7HZK6SBOSV6N7J","created_at":"2026-07-05T01:51:19Z"},{"alias_kind":"pith_short_8","alias_value":"3N7HZK6S","created_at":"2026-07-05T01:51:19Z"}],"graph_snapshots":[{"event_id":"sha256:3ab6f1f73960f48615a4018019c68a5be8d79bfbf79ad006c3180a87c0d46445","target":"graph","created_at":"2026-07-05T01:51:19Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2006.02066/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove new results on upper and lower limits of real-valued functions by means of $\\psi$-densities introduced by P. D. Barry in 1962. This allows us to improve several existing results on the growth of non-decreasing and unbounded real-valued functions in sets of positive density. The $\\psi$-densities are also used to introduce a new concept of a limit for real-valued functions. The results in this paper are of interest in real analysis as well as in the theory of meromorphic functions.","authors_text":"J. Heittokangas, J. Wang, M. A. Zemirni, Z. Latreuch","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2020-06-03T06:41:23Z","title":"On the limits of real-valued functions in sets involving $ \\psi $-density, and applications"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.02066","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:d82053b5c89f97c45ee029b542435fb464f773ff7f88ab19888bedf81e7dae40","target":"record","created_at":"2026-07-05T01:51:19Z","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":"daed14782ebb7f3bd95e64d69a8eed3f3709642ef0769d108dab7c1e8cbd81a7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2020-06-03T06:41:23Z","title_canon_sha256":"649fe671b7aad99d67efb9184e7e55a786364c60242106c659a92edab95b6950"},"schema_version":"1.0","source":{"id":"2006.02066","kind":"arxiv","version":2}},"canonical_sha256":"db7e7cabd20ba55f37e959509ff99eed6002107f90e04187f24de1779a2fc0bd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"db7e7cabd20ba55f37e959509ff99eed6002107f90e04187f24de1779a2fc0bd","first_computed_at":"2026-07-05T01:51:19.250977Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:51:19.250977Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mV/Czs0VhuOQY4NDXZJDTK0aRjsG7YU3tsP90sth+vC6/5oySt4SVgYUkYhWW5o7fGoQ/KC1/E0kW9fGrCN4Bg==","signature_status":"signed_v1","signed_at":"2026-07-05T01:51:19.251400Z","signed_message":"canonical_sha256_bytes"},"source_id":"2006.02066","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d82053b5c89f97c45ee029b542435fb464f773ff7f88ab19888bedf81e7dae40","sha256:3ab6f1f73960f48615a4018019c68a5be8d79bfbf79ad006c3180a87c0d46445"],"state_sha256":"3d2d2b313a15584a36afb7512b4f9d462b76582325485600c63b4264b9ae0f9a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MaHhUDl+izkpv4wIolfcvyS0OS7UTJ8UbH60nJ/vXVDM6rUHW/Tg23xxbmA3W+VU13V/8ZgoRYMj56TRQgdnAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-07T16:01:02.921140Z","bundle_sha256":"fad2c77903b73400943843a9813d4c1f0dba7583f6425da12eb2ffd36de719b5"}}