{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2014:CDWFOC2KNQVQYYWG5XOZIWDCRL","short_pith_number":"pith:CDWFOC2K","schema_version":"1.0","canonical_sha256":"10ec570b4a6c2b0c62c6eddd9458628aebf6b9ad481237cc1185f0a44d533072","source":{"kind":"arxiv","id":"1411.6868","version":1},"attestation_state":"computed","paper":{"title":"Bisector energy and few distinct distances","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adam Sheffer, Ben Lund, Frank de Zeeuw","submitted_at":"2014-11-25T13:51:36Z","abstract_excerpt":"We introduce the bisector energy of an $n$-point set $P$ in $\\mathbb{R}^2$, defined as the number of quadruples $(a,b,c,d)$ from $P$ such that $a$ and $b$ determine the same perpendicular bisector as $c$ and $d$. If no line or circle contains $M(n)$ points of $P$, then we prove that the bisector energy is $O(M(n)^{\\frac{2}{5}}n^{\\frac{12}{5}+\\epsilon} + M(n)n^2).$. We also prove the lower bound $\\Omega(M(n)n^2)$, which matches our upper bound when $M(n)$ is large. We use our upper bound on the bisector energy to obtain two rather different results:\n  (i) If $P$ determines $O(n/\\sqrt{\\log n})$ "},"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":"1411.6868","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-11-25T13:51:36Z","cross_cats_sorted":[],"title_canon_sha256":"fa810ce90960968c65664084b07db9d5c6c2020ef3f402e50a1e5112eeafc0cb","abstract_canon_sha256":"4e4e2372ec432f36a23dd0ca0c7fb28284d58aa049ad59cade350ff220f952f8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:32:47.108553Z","signature_b64":"jfbwDbo9wtLBXZL2/Zqr5/1d0/IJhPfXKlnYYZbY1IfqWW/IuS1l8B4vHaxukhJtfIjSv1g6Jx0AEeCpfKVnBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"10ec570b4a6c2b0c62c6eddd9458628aebf6b9ad481237cc1185f0a44d533072","last_reissued_at":"2026-05-18T02:32:47.108029Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:32:47.108029Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Bisector energy and few distinct distances","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adam Sheffer, Ben Lund, Frank de Zeeuw","submitted_at":"2014-11-25T13:51:36Z","abstract_excerpt":"We introduce the bisector energy of an $n$-point set $P$ in $\\mathbb{R}^2$, defined as the number of quadruples $(a,b,c,d)$ from $P$ such that $a$ and $b$ determine the same perpendicular bisector as $c$ and $d$. If no line or circle contains $M(n)$ points of $P$, then we prove that the bisector energy is $O(M(n)^{\\frac{2}{5}}n^{\\frac{12}{5}+\\epsilon} + M(n)n^2).$. We also prove the lower bound $\\Omega(M(n)n^2)$, which matches our upper bound when $M(n)$ is large. We use our upper bound on the bisector energy to obtain two rather different results:\n  (i) If $P$ determines $O(n/\\sqrt{\\log n})$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.6868","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":"1411.6868","created_at":"2026-05-18T02:32:47.108110+00:00"},{"alias_kind":"arxiv_version","alias_value":"1411.6868v1","created_at":"2026-05-18T02:32:47.108110+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.6868","created_at":"2026-05-18T02:32:47.108110+00:00"},{"alias_kind":"pith_short_12","alias_value":"CDWFOC2KNQVQ","created_at":"2026-05-18T12:28:22.404517+00:00"},{"alias_kind":"pith_short_16","alias_value":"CDWFOC2KNQVQYYWG","created_at":"2026-05-18T12:28:22.404517+00:00"},{"alias_kind":"pith_short_8","alias_value":"CDWFOC2K","created_at":"2026-05-18T12:28:22.404517+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/CDWFOC2KNQVQYYWG5XOZIWDCRL","json":"https://pith.science/pith/CDWFOC2KNQVQYYWG5XOZIWDCRL.json","graph_json":"https://pith.science/api/pith-number/CDWFOC2KNQVQYYWG5XOZIWDCRL/graph.json","events_json":"https://pith.science/api/pith-number/CDWFOC2KNQVQYYWG5XOZIWDCRL/events.json","paper":"https://pith.science/paper/CDWFOC2K"},"agent_actions":{"view_html":"https://pith.science/pith/CDWFOC2KNQVQYYWG5XOZIWDCRL","download_json":"https://pith.science/pith/CDWFOC2KNQVQYYWG5XOZIWDCRL.json","view_paper":"https://pith.science/paper/CDWFOC2K","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1411.6868&json=true","fetch_graph":"https://pith.science/api/pith-number/CDWFOC2KNQVQYYWG5XOZIWDCRL/graph.json","fetch_events":"https://pith.science/api/pith-number/CDWFOC2KNQVQYYWG5XOZIWDCRL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CDWFOC2KNQVQYYWG5XOZIWDCRL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CDWFOC2KNQVQYYWG5XOZIWDCRL/action/storage_attestation","attest_author":"https://pith.science/pith/CDWFOC2KNQVQYYWG5XOZIWDCRL/action/author_attestation","sign_citation":"https://pith.science/pith/CDWFOC2KNQVQYYWG5XOZIWDCRL/action/citation_signature","submit_replication":"https://pith.science/pith/CDWFOC2KNQVQYYWG5XOZIWDCRL/action/replication_record"}},"created_at":"2026-05-18T02:32:47.108110+00:00","updated_at":"2026-05-18T02:32:47.108110+00:00"}