{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:BY4ZFBJNLYMXBN6MX25SKMBORZ","short_pith_number":"pith:BY4ZFBJN","schema_version":"1.0","canonical_sha256":"0e3992852d5e1970b7ccbebb25302e8e4981e3f78386fbe5c91b8ac0147a7a52","source":{"kind":"arxiv","id":"2507.15679","version":1},"attestation_state":"computed","paper":{"title":"Erd\\H{o}s's unit distance problem and rigidity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"J\\'anos Pach, J\\'ozsef Solymosi, Orit E. Raz","submitted_at":"2025-07-21T14:43:01Z","abstract_excerpt":"According to a classical result of Spencer, Szemer\\'edi, and Trotter (1984), the maximum number of times the unit distance can occur among $n$ points in the plane is $O(n^{4/3})$. This is far from Erd\\H{o}s's lower bound, $n^{1+O(1/\\log\\log n)}$, which is conjectured to be optimal. We prove a structural result for point sets with nearly $n^{4/3}$ unit distances and use it to reduce the problem to a conjecture on rigid frameworks. This conjecture, if true, would yield the first improvement on the bound of Spencer et al. A weaker version of this conjecture has been established by the last two au"},"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":"2507.15679","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-07-21T14:43:01Z","cross_cats_sorted":[],"title_canon_sha256":"8cb7a22b71cff39ca120883b9e33ac51a1b4d0352b504207332efb09dbd2232b","abstract_canon_sha256":"8eb27b88b805e102e0ff21c854ecc89162cc7f39eaf3705b6665526fbada2212"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:40:41.882842Z","signature_b64":"lEu4V84yxseTH7wK0EI++Q6dHOhJDOL7Fi1xTJvbhM8PtfK+6ftxZtjt0iVPEZJQpnupvqSZ7fcEOGXjW/f8Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0e3992852d5e1970b7ccbebb25302e8e4981e3f78386fbe5c91b8ac0147a7a52","last_reissued_at":"2026-07-05T11:40:41.882334Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:40:41.882334Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Erd\\H{o}s's unit distance problem and rigidity","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"J\\'anos Pach, J\\'ozsef Solymosi, Orit E. Raz","submitted_at":"2025-07-21T14:43:01Z","abstract_excerpt":"According to a classical result of Spencer, Szemer\\'edi, and Trotter (1984), the maximum number of times the unit distance can occur among $n$ points in the plane is $O(n^{4/3})$. This is far from Erd\\H{o}s's lower bound, $n^{1+O(1/\\log\\log n)}$, which is conjectured to be optimal. We prove a structural result for point sets with nearly $n^{4/3}$ unit distances and use it to reduce the problem to a conjecture on rigid frameworks. This conjecture, if true, would yield the first improvement on the bound of Spencer et al. A weaker version of this conjecture has been established by the last two au"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.15679","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.15679/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":"2507.15679","created_at":"2026-07-05T11:40:41.882391+00:00"},{"alias_kind":"arxiv_version","alias_value":"2507.15679v1","created_at":"2026-07-05T11:40:41.882391+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.15679","created_at":"2026-07-05T11:40:41.882391+00:00"},{"alias_kind":"pith_short_12","alias_value":"BY4ZFBJNLYMX","created_at":"2026-07-05T11:40:41.882391+00:00"},{"alias_kind":"pith_short_16","alias_value":"BY4ZFBJNLYMXBN6M","created_at":"2026-07-05T11:40:41.882391+00:00"},{"alias_kind":"pith_short_8","alias_value":"BY4ZFBJN","created_at":"2026-07-05T11:40:41.882391+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/BY4ZFBJNLYMXBN6MX25SKMBORZ","json":"https://pith.science/pith/BY4ZFBJNLYMXBN6MX25SKMBORZ.json","graph_json":"https://pith.science/api/pith-number/BY4ZFBJNLYMXBN6MX25SKMBORZ/graph.json","events_json":"https://pith.science/api/pith-number/BY4ZFBJNLYMXBN6MX25SKMBORZ/events.json","paper":"https://pith.science/paper/BY4ZFBJN"},"agent_actions":{"view_html":"https://pith.science/pith/BY4ZFBJNLYMXBN6MX25SKMBORZ","download_json":"https://pith.science/pith/BY4ZFBJNLYMXBN6MX25SKMBORZ.json","view_paper":"https://pith.science/paper/BY4ZFBJN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2507.15679&json=true","fetch_graph":"https://pith.science/api/pith-number/BY4ZFBJNLYMXBN6MX25SKMBORZ/graph.json","fetch_events":"https://pith.science/api/pith-number/BY4ZFBJNLYMXBN6MX25SKMBORZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BY4ZFBJNLYMXBN6MX25SKMBORZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BY4ZFBJNLYMXBN6MX25SKMBORZ/action/storage_attestation","attest_author":"https://pith.science/pith/BY4ZFBJNLYMXBN6MX25SKMBORZ/action/author_attestation","sign_citation":"https://pith.science/pith/BY4ZFBJNLYMXBN6MX25SKMBORZ/action/citation_signature","submit_replication":"https://pith.science/pith/BY4ZFBJNLYMXBN6MX25SKMBORZ/action/replication_record"}},"created_at":"2026-07-05T11:40:41.882391+00:00","updated_at":"2026-07-05T11:40:41.882391+00:00"}