{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:KAF2PEM42NOJ547PHYXYLTIG6M","short_pith_number":"pith:KAF2PEM4","canonical_record":{"source":{"id":"1612.08008","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-12-23T15:41:18Z","cross_cats_sorted":[],"title_canon_sha256":"7305a243e467c594c05766153d7b92c8d73101c8ea3747af4e8630ba114d2fe7","abstract_canon_sha256":"97efae04bf7d80253386d10d37ff412af74c0a877a473886a487d555ece865a8"},"schema_version":"1.0"},"canonical_sha256":"500ba7919cd35c9ef3ef3e2f85cd06f31ef176cbff3c5089091a93c1713653c6","source":{"kind":"arxiv","id":"1612.08008","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.08008","created_at":"2026-05-18T00:42:07Z"},{"alias_kind":"arxiv_version","alias_value":"1612.08008v2","created_at":"2026-05-18T00:42:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.08008","created_at":"2026-05-18T00:42:07Z"},{"alias_kind":"pith_short_12","alias_value":"KAF2PEM42NOJ","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"KAF2PEM42NOJ547P","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"KAF2PEM4","created_at":"2026-05-18T12:30:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:KAF2PEM42NOJ547PHYXYLTIG6M","target":"record","payload":{"canonical_record":{"source":{"id":"1612.08008","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-12-23T15:41:18Z","cross_cats_sorted":[],"title_canon_sha256":"7305a243e467c594c05766153d7b92c8d73101c8ea3747af4e8630ba114d2fe7","abstract_canon_sha256":"97efae04bf7d80253386d10d37ff412af74c0a877a473886a487d555ece865a8"},"schema_version":"1.0"},"canonical_sha256":"500ba7919cd35c9ef3ef3e2f85cd06f31ef176cbff3c5089091a93c1713653c6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:07.277589Z","signature_b64":"URKwfO5QFuDmFK3F2ue4Wro6p4U2p4UQ20adgSN+VHSfZyFjPFnEjRkt/6yKwudleVJZxDtf3GNIoCNqMvqlAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"500ba7919cd35c9ef3ef3e2f85cd06f31ef176cbff3c5089091a93c1713653c6","last_reissued_at":"2026-05-18T00:42:07.276858Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:07.276858Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1612.08008","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":"2026-05-18T00:42:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"B3qce9yWpLlEL7DAAIvZw1IjnEDeGFQY5ZnT+7zY3QkB5xmhy1w3z8S/b1GKD2rAOgYNx8+TWKztZtsQ89aCDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T05:07:22.856875Z"},"content_sha256":"2ee683e2363aba7d6dd1293498c0765fe91334ba8f4f68f7de6e1db61c3c2ea6","schema_version":"1.0","event_id":"sha256:2ee683e2363aba7d6dd1293498c0765fe91334ba8f4f68f7de6e1db61c3c2ea6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:KAF2PEM42NOJ547PHYXYLTIG6M","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On pair correlation and discrepancy","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Gerhard Larcher, Sigrid Grepstad","submitted_at":"2016-12-23T15:41:18Z","abstract_excerpt":"We say that a sequence $\\{x_n\\}_{n \\geq 1}$ in $[0,1)$ has Poissonian pair correlations if\n  \\begin{equation*}\n  \\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\# \\left\\{ 1 \\leq l \\neq m \\leq N \\, : \\, \\left\\lVert x_l-x_m \\right\\rVert < \\frac{s}{N} \\right\\} = 2s\n  \\end{equation*} for all $s>0$. In this note we show that if the convergence in the above expression is - in a certain sense - fast, then this implies a small discrepancy for the sequence $\\{x_n\\}_{n \\geq 1}$. As an easy consequence it follows that every sequence with Poissonian pair correlations is uniformly distributed in $[0,1)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.08008","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":"2026-05-18T00:42:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u6NeTDxin/cJkzvLJNC1ph2cwnkReSS6H+fjHhjEzVdj/GYN6UDKXsTO1YY6Ef/FZJdbKoYK5N4AAoZa36meBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-20T05:07:22.857218Z"},"content_sha256":"01214ad8ecf6b52ce02012dae8e4f4f85686a86541c7ce6a67e0e32a233e5df0","schema_version":"1.0","event_id":"sha256:01214ad8ecf6b52ce02012dae8e4f4f85686a86541c7ce6a67e0e32a233e5df0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/KAF2PEM42NOJ547PHYXYLTIG6M/bundle.json","state_url":"https://pith.science/pith/KAF2PEM42NOJ547PHYXYLTIG6M/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/KAF2PEM42NOJ547PHYXYLTIG6M/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":"2026-06-20T05:07:22Z","links":{"resolver":"https://pith.science/pith/KAF2PEM42NOJ547PHYXYLTIG6M","bundle":"https://pith.science/pith/KAF2PEM42NOJ547PHYXYLTIG6M/bundle.json","state":"https://pith.science/pith/KAF2PEM42NOJ547PHYXYLTIG6M/state.json","well_known_bundle":"https://pith.science/.well-known/pith/KAF2PEM42NOJ547PHYXYLTIG6M/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:KAF2PEM42NOJ547PHYXYLTIG6M","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":"97efae04bf7d80253386d10d37ff412af74c0a877a473886a487d555ece865a8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-12-23T15:41:18Z","title_canon_sha256":"7305a243e467c594c05766153d7b92c8d73101c8ea3747af4e8630ba114d2fe7"},"schema_version":"1.0","source":{"id":"1612.08008","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.08008","created_at":"2026-05-18T00:42:07Z"},{"alias_kind":"arxiv_version","alias_value":"1612.08008v2","created_at":"2026-05-18T00:42:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.08008","created_at":"2026-05-18T00:42:07Z"},{"alias_kind":"pith_short_12","alias_value":"KAF2PEM42NOJ","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_16","alias_value":"KAF2PEM42NOJ547P","created_at":"2026-05-18T12:30:25Z"},{"alias_kind":"pith_short_8","alias_value":"KAF2PEM4","created_at":"2026-05-18T12:30:25Z"}],"graph_snapshots":[{"event_id":"sha256:01214ad8ecf6b52ce02012dae8e4f4f85686a86541c7ce6a67e0e32a233e5df0","target":"graph","created_at":"2026-05-18T00:42:07Z","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 say that a sequence $\\{x_n\\}_{n \\geq 1}$ in $[0,1)$ has Poissonian pair correlations if\n  \\begin{equation*}\n  \\lim_{N \\rightarrow \\infty} \\frac{1}{N} \\# \\left\\{ 1 \\leq l \\neq m \\leq N \\, : \\, \\left\\lVert x_l-x_m \\right\\rVert < \\frac{s}{N} \\right\\} = 2s\n  \\end{equation*} for all $s>0$. In this note we show that if the convergence in the above expression is - in a certain sense - fast, then this implies a small discrepancy for the sequence $\\{x_n\\}_{n \\geq 1}$. As an easy consequence it follows that every sequence with Poissonian pair correlations is uniformly distributed in $[0,1)$.","authors_text":"Gerhard Larcher, Sigrid Grepstad","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-12-23T15:41:18Z","title":"On pair correlation and discrepancy"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.08008","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":"sha256:2ee683e2363aba7d6dd1293498c0765fe91334ba8f4f68f7de6e1db61c3c2ea6","target":"record","created_at":"2026-05-18T00:42:07Z","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":"97efae04bf7d80253386d10d37ff412af74c0a877a473886a487d555ece865a8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-12-23T15:41:18Z","title_canon_sha256":"7305a243e467c594c05766153d7b92c8d73101c8ea3747af4e8630ba114d2fe7"},"schema_version":"1.0","source":{"id":"1612.08008","kind":"arxiv","version":2}},"canonical_sha256":"500ba7919cd35c9ef3ef3e2f85cd06f31ef176cbff3c5089091a93c1713653c6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"500ba7919cd35c9ef3ef3e2f85cd06f31ef176cbff3c5089091a93c1713653c6","first_computed_at":"2026-05-18T00:42:07.276858Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:42:07.276858Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"URKwfO5QFuDmFK3F2ue4Wro6p4U2p4UQ20adgSN+VHSfZyFjPFnEjRkt/6yKwudleVJZxDtf3GNIoCNqMvqlAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:42:07.277589Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.08008","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2ee683e2363aba7d6dd1293498c0765fe91334ba8f4f68f7de6e1db61c3c2ea6","sha256:01214ad8ecf6b52ce02012dae8e4f4f85686a86541c7ce6a67e0e32a233e5df0"],"state_sha256":"b02b2aa09edc42c95b2aa1d82c388f8a894f1d17f9fc8c599acac58f0cb1ab52"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lzdLR/2QSGDyvDSTtT94xZdz3hSCCIIhOfwjSHpnF77WKu7V+tR1o0pMIekD93EHpU1BdWpgTI8rpHa2mS8vCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-20T05:07:22.859202Z","bundle_sha256":"4554a6f9ebb6ab09c76debf4be3477483170d10b1cf766f055cfcd7b1283e8fc"}}