{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:JADOHYKCILLH7L4KLK6CU7QDXM","short_pith_number":"pith:JADOHYKC","schema_version":"1.0","canonical_sha256":"4806e3e14242d67faf8a5abc2a7e03bb0e0681a422d88103a11e86ced64eb78b","source":{"kind":"arxiv","id":"2501.11156","version":3},"attestation_state":"computed","paper":{"title":"Covering half-grids with lines and planes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"Anurag Bishnoi, Shantanu Nene","submitted_at":"2025-01-19T20:01:04Z","abstract_excerpt":"We study hyperplane covering problems for finite grid-like structures in $\\mathbb{R}^d$. We call a set $\\mathcal{C}$ of points in $\\mathbb{R}^2$ a conical grid if the line $y = a_i$ intersects $\\mathcal{C}$ in exactly $i$ points, for some $a_1 > \\cdots > a_n \\in \\mathbb{R}$. We prove that the number of lines required to cover every point of such a grid at least $k$ times is at least $nk\\left(1-\\frac{1}{e}-O(\\frac{1}{n}) \\right)$. If the grid $\\mathcal{C}$ is obtained by cutting an $m \\times n$ grid of points in half along one of the diagonals, then we prove the lower bound of $mk\\left(1-e^{-\\f"},"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":"2501.11156","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-01-19T20:01:04Z","cross_cats_sorted":["cs.CG"],"title_canon_sha256":"77f4e6bcc2595afb92a45b1cc86e1c645deb25cb6775ffd79f20f684157f1131","abstract_canon_sha256":"79620f89bf0aa146ee53822a9381d1d9ca7920beb0477a87ead3fc724722f500"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-12T01:09:06.856701Z","signature_b64":"5HpY+E2VClNOInBw81WtXjE3XzEIrTA7QOmriq06xFz3kUYMIuu+gYjvijCDKiAX4wfUQZKhXnn7BQk+B2SvAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4806e3e14242d67faf8a5abc2a7e03bb0e0681a422d88103a11e86ced64eb78b","last_reissued_at":"2026-06-12T01:09:06.855683Z","signature_status":"signed_v1","first_computed_at":"2026-06-12T01:09:06.855683Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Covering half-grids with lines and planes","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"Anurag Bishnoi, Shantanu Nene","submitted_at":"2025-01-19T20:01:04Z","abstract_excerpt":"We study hyperplane covering problems for finite grid-like structures in $\\mathbb{R}^d$. We call a set $\\mathcal{C}$ of points in $\\mathbb{R}^2$ a conical grid if the line $y = a_i$ intersects $\\mathcal{C}$ in exactly $i$ points, for some $a_1 > \\cdots > a_n \\in \\mathbb{R}$. We prove that the number of lines required to cover every point of such a grid at least $k$ times is at least $nk\\left(1-\\frac{1}{e}-O(\\frac{1}{n}) \\right)$. If the grid $\\mathcal{C}$ is obtained by cutting an $m \\times n$ grid of points in half along one of the diagonals, then we prove the lower bound of $mk\\left(1-e^{-\\f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.11156","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2501.11156/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":"2501.11156","created_at":"2026-06-12T01:09:06.855832+00:00"},{"alias_kind":"arxiv_version","alias_value":"2501.11156v3","created_at":"2026-06-12T01:09:06.855832+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2501.11156","created_at":"2026-06-12T01:09:06.855832+00:00"},{"alias_kind":"pith_short_12","alias_value":"JADOHYKCILLH","created_at":"2026-06-12T01:09:06.855832+00:00"},{"alias_kind":"pith_short_16","alias_value":"JADOHYKCILLH7L4K","created_at":"2026-06-12T01:09:06.855832+00:00"},{"alias_kind":"pith_short_8","alias_value":"JADOHYKC","created_at":"2026-06-12T01:09:06.855832+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/JADOHYKCILLH7L4KLK6CU7QDXM","json":"https://pith.science/pith/JADOHYKCILLH7L4KLK6CU7QDXM.json","graph_json":"https://pith.science/api/pith-number/JADOHYKCILLH7L4KLK6CU7QDXM/graph.json","events_json":"https://pith.science/api/pith-number/JADOHYKCILLH7L4KLK6CU7QDXM/events.json","paper":"https://pith.science/paper/JADOHYKC"},"agent_actions":{"view_html":"https://pith.science/pith/JADOHYKCILLH7L4KLK6CU7QDXM","download_json":"https://pith.science/pith/JADOHYKCILLH7L4KLK6CU7QDXM.json","view_paper":"https://pith.science/paper/JADOHYKC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2501.11156&json=true","fetch_graph":"https://pith.science/api/pith-number/JADOHYKCILLH7L4KLK6CU7QDXM/graph.json","fetch_events":"https://pith.science/api/pith-number/JADOHYKCILLH7L4KLK6CU7QDXM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JADOHYKCILLH7L4KLK6CU7QDXM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JADOHYKCILLH7L4KLK6CU7QDXM/action/storage_attestation","attest_author":"https://pith.science/pith/JADOHYKCILLH7L4KLK6CU7QDXM/action/author_attestation","sign_citation":"https://pith.science/pith/JADOHYKCILLH7L4KLK6CU7QDXM/action/citation_signature","submit_replication":"https://pith.science/pith/JADOHYKCILLH7L4KLK6CU7QDXM/action/replication_record"}},"created_at":"2026-06-12T01:09:06.855832+00:00","updated_at":"2026-06-12T01:09:06.855832+00:00"}