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A sequence $(x_n)$ is said to have Poissonian pair correlation if, for all $s>0$,\n  $$ \\lim_{N \\rightarrow \\infty}{ \\frac{1}{N} \\# \\left\\{ 1 \\leq m \\neq n \\leq N: |x_m - x_n| \\leq \\frac{s}{N} \\right\\}} = 2s.$$ It is known that this implies uniform distribution of the sequence $(x_n)$. Hinrichs, Kaltenb\\\"ock, Larcher, Stockinger \\& Ullrich extended this result to higher dimensions and showed that sequences $(x_n)$ in $[0,1]^d$ that satisfy, for all $s>0$,\n  $$ \\lim_{N \\rightarrow \\infty}{ \\frac{1}{N} \\#"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1812.10458","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-12-26T18:49:15Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"c4dd3f7de22cc240b43140567dbad18e0197f10ec4821e3f387b01a571ab318f","abstract_canon_sha256":"b387fd97bbe2f71bf43887e0816729981f758d0f17dd13c71efb65a7b7e792af"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:40:41.301178Z","signature_b64":"fJ1Df+BT4O7pYmRCIa/xSIK+7k3nvUi80d4NxMc3FYV4pJn08DSaxdPNj8b0aco0QZ0LLGo3F8j25sprCtMcAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"40a31e892972cad00a4a985bc28195d9613ac6b01bde419c74f26fd78d6afd2e","last_reissued_at":"2026-05-17T23:40:41.300502Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:40:41.300502Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Poissonian Pair Correlation in Higher Dimensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CA","authors_text":"Stefan Steinerberger","submitted_at":"2018-12-26T18:49:15Z","abstract_excerpt":"Let $(x_n)_{n=1}^{\\infty}$ be a sequence on the torus $\\mathbb{T}$ (normalized to length 1). A sequence $(x_n)$ is said to have Poissonian pair correlation if, for all $s>0$,\n  $$ \\lim_{N \\rightarrow \\infty}{ \\frac{1}{N} \\# \\left\\{ 1 \\leq m \\neq n \\leq N: |x_m - x_n| \\leq \\frac{s}{N} \\right\\}} = 2s.$$ It is known that this implies uniform distribution of the sequence $(x_n)$. Hinrichs, Kaltenb\\\"ock, Larcher, Stockinger \\& Ullrich extended this result to higher dimensions and showed that sequences $(x_n)$ in $[0,1]^d$ that satisfy, for all $s>0$,\n  $$ \\lim_{N \\rightarrow \\infty}{ \\frac{1}{N} \\#"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.10458","kind":"arxiv","version":3},"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":"1812.10458","created_at":"2026-05-17T23:40:41.300624+00:00"},{"alias_kind":"arxiv_version","alias_value":"1812.10458v3","created_at":"2026-05-17T23:40:41.300624+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1812.10458","created_at":"2026-05-17T23:40:41.300624+00:00"},{"alias_kind":"pith_short_12","alias_value":"ICRR5CJJOLFN","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_16","alias_value":"ICRR5CJJOLFNACSK","created_at":"2026-05-18T12:32:28.185984+00:00"},{"alias_kind":"pith_short_8","alias_value":"ICRR5CJJ","created_at":"2026-05-18T12:32:28.185984+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"1908.06292","citing_title":"On the number of gaps of sequences with Poissonian Pair Correlations","ref_index":14,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ICRR5CJJOLFNACSKTBN4FAMV3F","json":"https://pith.science/pith/ICRR5CJJOLFNACSKTBN4FAMV3F.json","graph_json":"https://pith.science/api/pith-number/ICRR5CJJOLFNACSKTBN4FAMV3F/graph.json","events_json":"https://pith.science/api/pith-number/ICRR5CJJOLFNACSKTBN4FAMV3F/events.json","paper":"https://pith.science/paper/ICRR5CJJ"},"agent_actions":{"view_html":"https://pith.science/pith/ICRR5CJJOLFNACSKTBN4FAMV3F","download_json":"https://pith.science/pith/ICRR5CJJOLFNACSKTBN4FAMV3F.json","view_paper":"https://pith.science/paper/ICRR5CJJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1812.10458&json=true","fetch_graph":"https://pith.science/api/pith-number/ICRR5CJJOLFNACSKTBN4FAMV3F/graph.json","fetch_events":"https://pith.science/api/pith-number/ICRR5CJJOLFNACSKTBN4FAMV3F/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ICRR5CJJOLFNACSKTBN4FAMV3F/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ICRR5CJJOLFNACSKTBN4FAMV3F/action/storage_attestation","attest_author":"https://pith.science/pith/ICRR5CJJOLFNACSKTBN4FAMV3F/action/author_attestation","sign_citation":"https://pith.science/pith/ICRR5CJJOLFNACSKTBN4FAMV3F/action/citation_signature","submit_replication":"https://pith.science/pith/ICRR5CJJOLFNACSKTBN4FAMV3F/action/replication_record"}},"created_at":"2026-05-17T23:40:41.300624+00:00","updated_at":"2026-05-17T23:40:41.300624+00:00"}