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Let $(s_n)_{n\\in\\mathbb N}$ be any sequence of real numbers. We prove that as $n\\to\\infty$, the number of real zeros of $X_n$ in the interval $[s_n+a/n, s_n+ b/n]$ converges in distribution to the number of zeros in the interval $[a,b]$ of a stationary, zero-mean Gaussian proce"},"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":"1601.05740","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-01-21T18:35:26Z","cross_cats_sorted":[],"title_canon_sha256":"cf96b1f9a1a7730e6961b366ea831fc1925a4c224c886ebe94c475d14a086292","abstract_canon_sha256":"6af11bc0e4baebe0f47e123633e890f53d95c30e1176739487671ff98413c4b4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:14:52.555201Z","signature_b64":"H+/yuYa/G97Qo6M/q3jHEv+KfhFZSFq5BUXxi/igfZFCCQWO8YJjWQmHODRxYIXOb74iPOB9sousTKbwdcGEBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6d68e8d7e0971b12f360ab500c7d38ec21b9c47cd310916153ed3c77c5618715","last_reissued_at":"2026-05-18T01:14:52.554581Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:14:52.554581Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Local universality for real roots of random trigonometric polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Alexander Iksanov, Alexander Marynych, Zakhar Kabluchko","submitted_at":"2016-01-21T18:35:26Z","abstract_excerpt":"Consider a random trigonometric polynomial $X_n: \\mathbb R \\to \\mathbb R$ of the form $$ X_n(t) = \\sum_{k=1}^n \\left( \\xi_k \\sin (kt) + \\eta_k \\cos (kt)\\right), $$ where $(\\xi_1,\\eta_1),(\\xi_2,\\eta_2),\\ldots$ are independent identically distributed bivariate real random vectors with zero mean and unit covariance matrix. Let $(s_n)_{n\\in\\mathbb N}$ be any sequence of real numbers. We prove that as $n\\to\\infty$, the number of real zeros of $X_n$ in the interval $[s_n+a/n, s_n+ b/n]$ converges in distribution to the number of zeros in the interval $[a,b]$ of a stationary, zero-mean Gaussian proce"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.05740","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":"1601.05740","created_at":"2026-05-18T01:14:52.554664+00:00"},{"alias_kind":"arxiv_version","alias_value":"1601.05740v2","created_at":"2026-05-18T01:14:52.554664+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.05740","created_at":"2026-05-18T01:14:52.554664+00:00"},{"alias_kind":"pith_short_12","alias_value":"NVUORV7AS4NR","created_at":"2026-05-18T12:30:36.002864+00:00"},{"alias_kind":"pith_short_16","alias_value":"NVUORV7AS4NRF43A","created_at":"2026-05-18T12:30:36.002864+00:00"},{"alias_kind":"pith_short_8","alias_value":"NVUORV7A","created_at":"2026-05-18T12:30:36.002864+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/NVUORV7AS4NRF43AVNIAY7JY5Q","json":"https://pith.science/pith/NVUORV7AS4NRF43AVNIAY7JY5Q.json","graph_json":"https://pith.science/api/pith-number/NVUORV7AS4NRF43AVNIAY7JY5Q/graph.json","events_json":"https://pith.science/api/pith-number/NVUORV7AS4NRF43AVNIAY7JY5Q/events.json","paper":"https://pith.science/paper/NVUORV7A"},"agent_actions":{"view_html":"https://pith.science/pith/NVUORV7AS4NRF43AVNIAY7JY5Q","download_json":"https://pith.science/pith/NVUORV7AS4NRF43AVNIAY7JY5Q.json","view_paper":"https://pith.science/paper/NVUORV7A","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1601.05740&json=true","fetch_graph":"https://pith.science/api/pith-number/NVUORV7AS4NRF43AVNIAY7JY5Q/graph.json","fetch_events":"https://pith.science/api/pith-number/NVUORV7AS4NRF43AVNIAY7JY5Q/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NVUORV7AS4NRF43AVNIAY7JY5Q/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NVUORV7AS4NRF43AVNIAY7JY5Q/action/storage_attestation","attest_author":"https://pith.science/pith/NVUORV7AS4NRF43AVNIAY7JY5Q/action/author_attestation","sign_citation":"https://pith.science/pith/NVUORV7AS4NRF43AVNIAY7JY5Q/action/citation_signature","submit_replication":"https://pith.science/pith/NVUORV7AS4NRF43AVNIAY7JY5Q/action/replication_record"}},"created_at":"2026-05-18T01:14:52.554664+00:00","updated_at":"2026-05-18T01:14:52.554664+00:00"}