{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:744HIUVYAV7OP2NDFHOAQ6TBCP","short_pith_number":"pith:744HIUVY","schema_version":"1.0","canonical_sha256":"ff387452b8057ee7e9a329dc087a6113f9bb3e3549a1de3a56f733889c9e9118","source":{"kind":"arxiv","id":"1006.0743","version":1},"attestation_state":"computed","paper":{"title":"Bounds for solid angles of lattices of rank three","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"math.MG","authors_text":"Lenny Fukshansky, Sinai Robins","submitted_at":"2010-06-03T21:11:33Z","abstract_excerpt":"We find sharp absolute constants $C_1$ and $C_2$ with the following property: every well-rounded lattice of rank 3 in a Euclidean space has a minimal basis so that the solid angle spanned by these basis vectors lies in the interval $[C_1,C_2]$. In fact, we show that these absolute bounds hold for a larger class of lattices than just well-rounded, and the upper bound holds for all. We state a technical condition on the lattice that may prevent it from satisfying the absolute lower bound on the solid angle, in which case we derive a lower bound in terms of the ratios of successive minima of the "},"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":"1006.0743","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2010-06-03T21:11:33Z","cross_cats_sorted":["math.CO","math.NT"],"title_canon_sha256":"802d0763e233d5e79955391df5cddbc2a35e6aa27b3dd1090de64a991950f13c","abstract_canon_sha256":"f7e49307313d59b65cebf1338dec1116e643e8942d4669f4898066c18836ad3c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:34:46.013271Z","signature_b64":"EKlMJwMdgBBTnbDB3r8ZvDWCqWuDOUzPbJ0cxkyjKA0x5YXwgq1+cbrS0DILh5jQzwWjoZ8EcvDYIKAZ/iNKCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ff387452b8057ee7e9a329dc087a6113f9bb3e3549a1de3a56f733889c9e9118","last_reissued_at":"2026-05-18T04:34:46.012862Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:34:46.012862Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bounds for solid angles of lattices of rank three","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"math.MG","authors_text":"Lenny Fukshansky, Sinai Robins","submitted_at":"2010-06-03T21:11:33Z","abstract_excerpt":"We find sharp absolute constants $C_1$ and $C_2$ with the following property: every well-rounded lattice of rank 3 in a Euclidean space has a minimal basis so that the solid angle spanned by these basis vectors lies in the interval $[C_1,C_2]$. In fact, we show that these absolute bounds hold for a larger class of lattices than just well-rounded, and the upper bound holds for all. We state a technical condition on the lattice that may prevent it from satisfying the absolute lower bound on the solid angle, in which case we derive a lower bound in terms of the ratios of successive minima of the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.0743","kind":"arxiv","version":1},"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":"1006.0743","created_at":"2026-05-18T04:34:46.012918+00:00"},{"alias_kind":"arxiv_version","alias_value":"1006.0743v1","created_at":"2026-05-18T04:34:46.012918+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.0743","created_at":"2026-05-18T04:34:46.012918+00:00"},{"alias_kind":"pith_short_12","alias_value":"744HIUVYAV7O","created_at":"2026-05-18T12:26:04.259169+00:00"},{"alias_kind":"pith_short_16","alias_value":"744HIUVYAV7OP2ND","created_at":"2026-05-18T12:26:04.259169+00:00"},{"alias_kind":"pith_short_8","alias_value":"744HIUVY","created_at":"2026-05-18T12:26:04.259169+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/744HIUVYAV7OP2NDFHOAQ6TBCP","json":"https://pith.science/pith/744HIUVYAV7OP2NDFHOAQ6TBCP.json","graph_json":"https://pith.science/api/pith-number/744HIUVYAV7OP2NDFHOAQ6TBCP/graph.json","events_json":"https://pith.science/api/pith-number/744HIUVYAV7OP2NDFHOAQ6TBCP/events.json","paper":"https://pith.science/paper/744HIUVY"},"agent_actions":{"view_html":"https://pith.science/pith/744HIUVYAV7OP2NDFHOAQ6TBCP","download_json":"https://pith.science/pith/744HIUVYAV7OP2NDFHOAQ6TBCP.json","view_paper":"https://pith.science/paper/744HIUVY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1006.0743&json=true","fetch_graph":"https://pith.science/api/pith-number/744HIUVYAV7OP2NDFHOAQ6TBCP/graph.json","fetch_events":"https://pith.science/api/pith-number/744HIUVYAV7OP2NDFHOAQ6TBCP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/744HIUVYAV7OP2NDFHOAQ6TBCP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/744HIUVYAV7OP2NDFHOAQ6TBCP/action/storage_attestation","attest_author":"https://pith.science/pith/744HIUVYAV7OP2NDFHOAQ6TBCP/action/author_attestation","sign_citation":"https://pith.science/pith/744HIUVYAV7OP2NDFHOAQ6TBCP/action/citation_signature","submit_replication":"https://pith.science/pith/744HIUVYAV7OP2NDFHOAQ6TBCP/action/replication_record"}},"created_at":"2026-05-18T04:34:46.012918+00:00","updated_at":"2026-05-18T04:34:46.012918+00:00"}