{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:6O5GBHD77UHGKAT5HO5KXFY4SJ","short_pith_number":"pith:6O5GBHD7","schema_version":"1.0","canonical_sha256":"f3ba609c7ffd0e65027d3bbaab971c927cba5196164a1185b1fae09c641f2829","source":{"kind":"arxiv","id":"1306.0242","version":2},"attestation_state":"computed","paper":{"title":"On lattices, distinct distances, and the Elekes-Sharir framework","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"Adam Sheffer, Javier Cilleruelo, Micha Sharir","submitted_at":"2013-06-02T19:03:28Z","abstract_excerpt":"In this note we consider distinct distances determined by points in an integer lattice. We first consider Erdos's lower bound for the square lattice, recast in the setup of the so-called Elekes-Sharir framework \\cite{ES11,GK11}, and show that, without a major change, this framework \\emph{cannot} lead to Erdos's conjectured lower bound. This shows that the upper bound of Guth and Katz \\cite{GK11} for the related 3-dimensional line-intersection problem is tight for this instance. The gap between this bound and the actual bound of Erdos arises from an application of the Cauchy-Schwarz inequality "},"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":"1306.0242","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-06-02T19:03:28Z","cross_cats_sorted":["cs.CG"],"title_canon_sha256":"edb58d3b6f6a8b3655c86b12eb5fe909961932caee0e432a50a2c93a466a8dd6","abstract_canon_sha256":"b58ee943488a9fc73409137120ef924140240db958b2fa1cea3ce9803f5eb249"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:19:43.637117Z","signature_b64":"ob5I4qotgvjTiKQ6oCeed8iLUO1r969SNlHXPEJ5rjR/apw0Y7k8Ap/QyIhTGdmNfUky6PB2mPPolyb2UuXLAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f3ba609c7ffd0e65027d3bbaab971c927cba5196164a1185b1fae09c641f2829","last_reissued_at":"2026-05-18T03:19:43.636514Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:19:43.636514Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On lattices, distinct distances, and the Elekes-Sharir framework","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CG"],"primary_cat":"math.CO","authors_text":"Adam Sheffer, Javier Cilleruelo, Micha Sharir","submitted_at":"2013-06-02T19:03:28Z","abstract_excerpt":"In this note we consider distinct distances determined by points in an integer lattice. We first consider Erdos's lower bound for the square lattice, recast in the setup of the so-called Elekes-Sharir framework \\cite{ES11,GK11}, and show that, without a major change, this framework \\emph{cannot} lead to Erdos's conjectured lower bound. This shows that the upper bound of Guth and Katz \\cite{GK11} for the related 3-dimensional line-intersection problem is tight for this instance. The gap between this bound and the actual bound of Erdos arises from an application of the Cauchy-Schwarz inequality "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.0242","kind":"arxiv","version":2},"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":"1306.0242","created_at":"2026-05-18T03:19:43.636609+00:00"},{"alias_kind":"arxiv_version","alias_value":"1306.0242v2","created_at":"2026-05-18T03:19:43.636609+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1306.0242","created_at":"2026-05-18T03:19:43.636609+00:00"},{"alias_kind":"pith_short_12","alias_value":"6O5GBHD77UHG","created_at":"2026-05-18T12:27:36.564083+00:00"},{"alias_kind":"pith_short_16","alias_value":"6O5GBHD77UHGKAT5","created_at":"2026-05-18T12:27:36.564083+00:00"},{"alias_kind":"pith_short_8","alias_value":"6O5GBHD7","created_at":"2026-05-18T12:27:36.564083+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/6O5GBHD77UHGKAT5HO5KXFY4SJ","json":"https://pith.science/pith/6O5GBHD77UHGKAT5HO5KXFY4SJ.json","graph_json":"https://pith.science/api/pith-number/6O5GBHD77UHGKAT5HO5KXFY4SJ/graph.json","events_json":"https://pith.science/api/pith-number/6O5GBHD77UHGKAT5HO5KXFY4SJ/events.json","paper":"https://pith.science/paper/6O5GBHD7"},"agent_actions":{"view_html":"https://pith.science/pith/6O5GBHD77UHGKAT5HO5KXFY4SJ","download_json":"https://pith.science/pith/6O5GBHD77UHGKAT5HO5KXFY4SJ.json","view_paper":"https://pith.science/paper/6O5GBHD7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1306.0242&json=true","fetch_graph":"https://pith.science/api/pith-number/6O5GBHD77UHGKAT5HO5KXFY4SJ/graph.json","fetch_events":"https://pith.science/api/pith-number/6O5GBHD77UHGKAT5HO5KXFY4SJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6O5GBHD77UHGKAT5HO5KXFY4SJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6O5GBHD77UHGKAT5HO5KXFY4SJ/action/storage_attestation","attest_author":"https://pith.science/pith/6O5GBHD77UHGKAT5HO5KXFY4SJ/action/author_attestation","sign_citation":"https://pith.science/pith/6O5GBHD77UHGKAT5HO5KXFY4SJ/action/citation_signature","submit_replication":"https://pith.science/pith/6O5GBHD77UHGKAT5HO5KXFY4SJ/action/replication_record"}},"created_at":"2026-05-18T03:19:43.636609+00:00","updated_at":"2026-05-18T03:19:43.636609+00:00"}