{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:MOOQP5TWZ7LACFWBHM4LGAWDRD","short_pith_number":"pith:MOOQP5TW","schema_version":"1.0","canonical_sha256":"639d07f676cfd60116c13b38b302c388de1b8120771dfb63094b5f809df353d5","source":{"kind":"arxiv","id":"1412.5394","version":1},"attestation_state":"computed","paper":{"title":"Balancing Sets of Vectors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"G\\'abor Heged\\\"us","submitted_at":"2014-12-17T13:58:45Z","abstract_excerpt":"Let $n$ be an arbitrary integer, let $p$ be a prime factor of $n$. Denote by $\\omega_1$ the $p^{th}$ primitive unity root, $\\omega_1:=e^{\\frac{2\\pi i}{p}}$. Define $\\omega_i:=\\omega_1^i$ for $0\\leq i\\leq p-1$ and $B:=\\{1,\\omega_1,...,\\omega_{p-1}\\}^n$. Denote by $K(n,p)$ the minimum $k$ for which there exist vectors $v_1,...,v_k\\in B$ such that for any vector $w\\in B$, there is an $i$, $1\\leq i\\leq k$, such that $v_i\\cdot w=0$, where $v\\cdot w$ is the usual scalar product of $v$ and $w$. Gr\\\"obner basis methods and linear algebra proof gives the lower bound $K(n,p)\\geq n(p-1)$. Galvin posed th"},"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":"1412.5394","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-12-17T13:58:45Z","cross_cats_sorted":[],"title_canon_sha256":"d5a1805440eb9b8ca26d9dfe4e3385af214470af605bec006fc6dbd56eaf090a","abstract_canon_sha256":"39fda1e5a2f1b4089220f6aa3ee61e6700eb0354a62e6630d2c39d12a557fd2a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:02:59.536622Z","signature_b64":"w9InKa5xmmXhCeZMRe4iqstlU/Vbui20o0kDaB8PrgkmgWGp66lNeLIvP34KO1BrEqao5IhpPI+FICFK8Gt6Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"639d07f676cfd60116c13b38b302c388de1b8120771dfb63094b5f809df353d5","last_reissued_at":"2026-05-18T01:02:59.536050Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:02:59.536050Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Balancing Sets of Vectors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"G\\'abor Heged\\\"us","submitted_at":"2014-12-17T13:58:45Z","abstract_excerpt":"Let $n$ be an arbitrary integer, let $p$ be a prime factor of $n$. Denote by $\\omega_1$ the $p^{th}$ primitive unity root, $\\omega_1:=e^{\\frac{2\\pi i}{p}}$. Define $\\omega_i:=\\omega_1^i$ for $0\\leq i\\leq p-1$ and $B:=\\{1,\\omega_1,...,\\omega_{p-1}\\}^n$. Denote by $K(n,p)$ the minimum $k$ for which there exist vectors $v_1,...,v_k\\in B$ such that for any vector $w\\in B$, there is an $i$, $1\\leq i\\leq k$, such that $v_i\\cdot w=0$, where $v\\cdot w$ is the usual scalar product of $v$ and $w$. Gr\\\"obner basis methods and linear algebra proof gives the lower bound $K(n,p)\\geq n(p-1)$. Galvin posed th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.5394","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":"1412.5394","created_at":"2026-05-18T01:02:59.536121+00:00"},{"alias_kind":"arxiv_version","alias_value":"1412.5394v1","created_at":"2026-05-18T01:02:59.536121+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.5394","created_at":"2026-05-18T01:02:59.536121+00:00"},{"alias_kind":"pith_short_12","alias_value":"MOOQP5TWZ7LA","created_at":"2026-05-18T12:28:38.356838+00:00"},{"alias_kind":"pith_short_16","alias_value":"MOOQP5TWZ7LACFWB","created_at":"2026-05-18T12:28:38.356838+00:00"},{"alias_kind":"pith_short_8","alias_value":"MOOQP5TW","created_at":"2026-05-18T12:28:38.356838+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/MOOQP5TWZ7LACFWBHM4LGAWDRD","json":"https://pith.science/pith/MOOQP5TWZ7LACFWBHM4LGAWDRD.json","graph_json":"https://pith.science/api/pith-number/MOOQP5TWZ7LACFWBHM4LGAWDRD/graph.json","events_json":"https://pith.science/api/pith-number/MOOQP5TWZ7LACFWBHM4LGAWDRD/events.json","paper":"https://pith.science/paper/MOOQP5TW"},"agent_actions":{"view_html":"https://pith.science/pith/MOOQP5TWZ7LACFWBHM4LGAWDRD","download_json":"https://pith.science/pith/MOOQP5TWZ7LACFWBHM4LGAWDRD.json","view_paper":"https://pith.science/paper/MOOQP5TW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1412.5394&json=true","fetch_graph":"https://pith.science/api/pith-number/MOOQP5TWZ7LACFWBHM4LGAWDRD/graph.json","fetch_events":"https://pith.science/api/pith-number/MOOQP5TWZ7LACFWBHM4LGAWDRD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MOOQP5TWZ7LACFWBHM4LGAWDRD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MOOQP5TWZ7LACFWBHM4LGAWDRD/action/storage_attestation","attest_author":"https://pith.science/pith/MOOQP5TWZ7LACFWBHM4LGAWDRD/action/author_attestation","sign_citation":"https://pith.science/pith/MOOQP5TWZ7LACFWBHM4LGAWDRD/action/citation_signature","submit_replication":"https://pith.science/pith/MOOQP5TWZ7LACFWBHM4LGAWDRD/action/replication_record"}},"created_at":"2026-05-18T01:02:59.536121+00:00","updated_at":"2026-05-18T01:02:59.536121+00:00"}