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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 $[0,1)$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"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"},"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"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1612.08008","created_at":"2026-05-18T00:42:07.276962+00:00"},{"alias_kind":"arxiv_version","alias_value":"1612.08008v2","created_at":"2026-05-18T00:42:07.276962+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.08008","created_at":"2026-05-18T00:42:07.276962+00:00"},{"alias_kind":"pith_short_12","alias_value":"KAF2PEM42NOJ","created_at":"2026-05-18T12:30:25.849896+00:00"},{"alias_kind":"pith_short_16","alias_value":"KAF2PEM42NOJ547P","created_at":"2026-05-18T12:30:25.849896+00:00"},{"alias_kind":"pith_short_8","alias_value":"KAF2PEM4","created_at":"2026-05-18T12:30:25.849896+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/KAF2PEM42NOJ547PHYXYLTIG6M","json":"https://pith.science/pith/KAF2PEM42NOJ547PHYXYLTIG6M.json","graph_json":"https://pith.science/api/pith-number/KAF2PEM42NOJ547PHYXYLTIG6M/graph.json","events_json":"https://pith.science/api/pith-number/KAF2PEM42NOJ547PHYXYLTIG6M/events.json","paper":"https://pith.science/paper/KAF2PEM4"},"agent_actions":{"view_html":"https://pith.science/pith/KAF2PEM42NOJ547PHYXYLTIG6M","download_json":"https://pith.science/pith/KAF2PEM42NOJ547PHYXYLTIG6M.json","view_paper":"https://pith.science/paper/KAF2PEM4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1612.08008&json=true","fetch_graph":"https://pith.science/api/pith-number/KAF2PEM42NOJ547PHYXYLTIG6M/graph.json","fetch_events":"https://pith.science/api/pith-number/KAF2PEM42NOJ547PHYXYLTIG6M/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/KAF2PEM42NOJ547PHYXYLTIG6M/action/timestamp_anchor","attest_storage":"https://pith.science/pith/KAF2PEM42NOJ547PHYXYLTIG6M/action/storage_attestation","attest_author":"https://pith.science/pith/KAF2PEM42NOJ547PHYXYLTIG6M/action/author_attestation","sign_citation":"https://pith.science/pith/KAF2PEM42NOJ547PHYXYLTIG6M/action/citation_signature","submit_replication":"https://pith.science/pith/KAF2PEM42NOJ547PHYXYLTIG6M/action/replication_record"}},"created_at":"2026-05-18T00:42:07.276962+00:00","updated_at":"2026-05-18T00:42:07.276962+00:00"}