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We prove that for any $\\varepsilon>0$ and any $M \\in \\mathbb{N}$, there exists a universal set $\\Lambda \\subset \\mathbb{R}$ of upper Beurling density less than $1+\\varepsilon$ such that the system $$\\left\\{ e^{2\\pi i \\lambda t } g(t-n) \\colon (\\lambda, n) \\in \\Lambda \\times \\mathbb{Z} \\right\\}$$\n  forms a frame in $L^2(\\mathbb{R})$ for any well-behaved rational function $g$."},"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":"2510.25930","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2025-10-29T20:09:25Z","cross_cats_sorted":[],"title_canon_sha256":"de4db92b57f537d7e4c2c6e2df8fd606339d715b5e9c673eb79c40626a204d32","abstract_canon_sha256":"4d99f53227e9ebbb0b0f2aeefad6e89b2265f959be92ac7557e83828eb0ba75a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-07T02:17:12.645239Z","signature_b64":"tms9GQGC7aTofF32703OmGN0gQ5ZgFae6Yhq3ax8KCWU6+wr8rZMOUhY5/eVY2dIJn3fCPk8qTm1wRtzMPGlCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"210a4b9a62d8e33f7041ed7eb09940a319053aea7a990fe12603a4fb64fe45af","last_reissued_at":"2026-07-07T02:17:12.644054Z","signature_status":"signed_v1","first_computed_at":"2026-07-07T02:17:12.644054Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Universal frame set for rational functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Andrei V. 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We prove that for any $\\varepsilon>0$ and any $M \\in \\mathbb{N}$, there exists a universal set $\\Lambda \\subset \\mathbb{R}$ of upper Beurling density less than $1+\\varepsilon$ such that the system $$\\left\\{ e^{2\\pi i \\lambda t } g(t-n) \\colon (\\lambda, n) \\in \\Lambda \\times \\mathbb{Z} \\right\\}$$\n  forms a frame in $L^2(\\mathbb{R})$ for any well-behaved rational function $g$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2510.25930","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/2510.25930/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2510.25930","created_at":"2026-07-07T02:17:12.644214+00:00"},{"alias_kind":"arxiv_version","alias_value":"2510.25930v2","created_at":"2026-07-07T02:17:12.644214+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2510.25930","created_at":"2026-07-07T02:17:12.644214+00:00"},{"alias_kind":"pith_short_12","alias_value":"EEFEXGTC3DRT","created_at":"2026-07-07T02:17:12.644214+00:00"},{"alias_kind":"pith_short_16","alias_value":"EEFEXGTC3DRT64CB","created_at":"2026-07-07T02:17:12.644214+00:00"},{"alias_kind":"pith_short_8","alias_value":"EEFEXGTC","created_at":"2026-07-07T02:17:12.644214+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/EEFEXGTC3DRT64CB5V7LBGKAUM","json":"https://pith.science/pith/EEFEXGTC3DRT64CB5V7LBGKAUM.json","graph_json":"https://pith.science/api/pith-number/EEFEXGTC3DRT64CB5V7LBGKAUM/graph.json","events_json":"https://pith.science/api/pith-number/EEFEXGTC3DRT64CB5V7LBGKAUM/events.json","paper":"https://pith.science/paper/EEFEXGTC"},"agent_actions":{"view_html":"https://pith.science/pith/EEFEXGTC3DRT64CB5V7LBGKAUM","download_json":"https://pith.science/pith/EEFEXGTC3DRT64CB5V7LBGKAUM.json","view_paper":"https://pith.science/paper/EEFEXGTC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2510.25930&json=true","fetch_graph":"https://pith.science/api/pith-number/EEFEXGTC3DRT64CB5V7LBGKAUM/graph.json","fetch_events":"https://pith.science/api/pith-number/EEFEXGTC3DRT64CB5V7LBGKAUM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EEFEXGTC3DRT64CB5V7LBGKAUM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EEFEXGTC3DRT64CB5V7LBGKAUM/action/storage_attestation","attest_author":"https://pith.science/pith/EEFEXGTC3DRT64CB5V7LBGKAUM/action/author_attestation","sign_citation":"https://pith.science/pith/EEFEXGTC3DRT64CB5V7LBGKAUM/action/citation_signature","submit_replication":"https://pith.science/pith/EEFEXGTC3DRT64CB5V7LBGKAUM/action/replication_record"}},"created_at":"2026-07-07T02:17:12.644214+00:00","updated_at":"2026-07-07T02:17:12.644214+00:00"}