{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:42CBRBCIME7KKWXINXGRX6BWEK","short_pith_number":"pith:42CBRBCI","schema_version":"1.0","canonical_sha256":"e684188448613ea55ae86dcd1bf8362296138ab292f351896c736997ba9fe46f","source":{"kind":"arxiv","id":"1405.7657","version":1},"attestation_state":"computed","paper":{"title":"The square root law and structure of finite rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.CO"],"primary_cat":"math.NT","authors_text":"A. Iosevich, B. Murphy, J. Pakianathan","submitted_at":"2014-05-29T19:07:23Z","abstract_excerpt":"Let $R$ be a finite ring and define the hyperbola $H=\\{(x,y) \\in R \\times R: xy=1 \\}$. Suppose that for a sequence of finite odd order rings of size tending to infinity, the following \"square root law\" bound holds with a constant $C>0$ for all non-trivial characters $\\chi$ on $R^2$: \\[ \\left| \\sum_{(x,y)\\in H}\\chi(x,y)\\right|\\leq C\\sqrt{|H|}. \\] Then, with a finite number of exceptions, those rings are fields.\n  For rings of even order we show that there are other infinite families given by Boolean rings and Boolean twists which satisfy this square-root law behavior. We classify the extremal r"},"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":"1405.7657","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-05-29T19:07:23Z","cross_cats_sorted":["math.CA","math.CO"],"title_canon_sha256":"1feddab28dad4dbe49219cdc7826f97e6a72bd9d9d72a5a635143f45ca2b1e0e","abstract_canon_sha256":"b4d88b1343c9d384241481628c86288d2581d4eb10f63e4aea2f18b094b502bd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:50:50.781862Z","signature_b64":"fJdFizO1LntcneWCxZkQGeMgakEJbphgA9j4L+D+O7dTJZQsBOz7U1FEGuaiTk1HlEHq2BDjlD+y8z7oJR2AAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e684188448613ea55ae86dcd1bf8362296138ab292f351896c736997ba9fe46f","last_reissued_at":"2026-05-18T02:50:50.781372Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:50:50.781372Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The square root law and structure of finite rings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.CO"],"primary_cat":"math.NT","authors_text":"A. Iosevich, B. Murphy, J. Pakianathan","submitted_at":"2014-05-29T19:07:23Z","abstract_excerpt":"Let $R$ be a finite ring and define the hyperbola $H=\\{(x,y) \\in R \\times R: xy=1 \\}$. Suppose that for a sequence of finite odd order rings of size tending to infinity, the following \"square root law\" bound holds with a constant $C>0$ for all non-trivial characters $\\chi$ on $R^2$: \\[ \\left| \\sum_{(x,y)\\in H}\\chi(x,y)\\right|\\leq C\\sqrt{|H|}. \\] Then, with a finite number of exceptions, those rings are fields.\n  For rings of even order we show that there are other infinite families given by Boolean rings and Boolean twists which satisfy this square-root law behavior. We classify the extremal r"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.7657","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":"1405.7657","created_at":"2026-05-18T02:50:50.781463+00:00"},{"alias_kind":"arxiv_version","alias_value":"1405.7657v1","created_at":"2026-05-18T02:50:50.781463+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1405.7657","created_at":"2026-05-18T02:50:50.781463+00:00"},{"alias_kind":"pith_short_12","alias_value":"42CBRBCIME7K","created_at":"2026-05-18T12:28:11.866339+00:00"},{"alias_kind":"pith_short_16","alias_value":"42CBRBCIME7KKWXI","created_at":"2026-05-18T12:28:11.866339+00:00"},{"alias_kind":"pith_short_8","alias_value":"42CBRBCI","created_at":"2026-05-18T12:28:11.866339+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/42CBRBCIME7KKWXINXGRX6BWEK","json":"https://pith.science/pith/42CBRBCIME7KKWXINXGRX6BWEK.json","graph_json":"https://pith.science/api/pith-number/42CBRBCIME7KKWXINXGRX6BWEK/graph.json","events_json":"https://pith.science/api/pith-number/42CBRBCIME7KKWXINXGRX6BWEK/events.json","paper":"https://pith.science/paper/42CBRBCI"},"agent_actions":{"view_html":"https://pith.science/pith/42CBRBCIME7KKWXINXGRX6BWEK","download_json":"https://pith.science/pith/42CBRBCIME7KKWXINXGRX6BWEK.json","view_paper":"https://pith.science/paper/42CBRBCI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1405.7657&json=true","fetch_graph":"https://pith.science/api/pith-number/42CBRBCIME7KKWXINXGRX6BWEK/graph.json","fetch_events":"https://pith.science/api/pith-number/42CBRBCIME7KKWXINXGRX6BWEK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/42CBRBCIME7KKWXINXGRX6BWEK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/42CBRBCIME7KKWXINXGRX6BWEK/action/storage_attestation","attest_author":"https://pith.science/pith/42CBRBCIME7KKWXINXGRX6BWEK/action/author_attestation","sign_citation":"https://pith.science/pith/42CBRBCIME7KKWXINXGRX6BWEK/action/citation_signature","submit_replication":"https://pith.science/pith/42CBRBCIME7KKWXINXGRX6BWEK/action/replication_record"}},"created_at":"2026-05-18T02:50:50.781463+00:00","updated_at":"2026-05-18T02:50:50.781463+00:00"}