{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:YUHTKQZPZGPVFH36BEC6SMV4NH","short_pith_number":"pith:YUHTKQZP","schema_version":"1.0","canonical_sha256":"c50f35432fc99f529f7e0905e932bc69d27ff87cc25bd96dbd923a998d865527","source":{"kind":"arxiv","id":"2607.05374","version":1},"attestation_state":"computed","paper":{"title":"The Minkowski grid has robustly many repeated distances","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.NT"],"primary_cat":"math.CO","authors_text":"Cosmin Pohoata, Daniel G. Zhu, Sungchul Lee","submitted_at":"2026-07-06T17:52:52Z","abstract_excerpt":"We show that there exists a constant $\\delta > 0$ such that for any positive integer $n$ there exists a set of $n$ points $P \\subset \\mathbb{R}^2$ with the following property: for every subset $A \\subseteq P$ of size $|A| \\geq 2$, \\[ \\max_{\\lambda>0}\n  \\#\\{(a,b)\\in A \\times A:\n  a\\ne b,\\ \\lvert a-b\\rvert=\\lambda\\} \\gtrsim \\frac{|A|^2}{n^{1-\\delta}}.\\] Our result is a vertical amplification of a robust Ramanujan estimate recently established by Croot-Mao-Pohoata-Sheffer-Yip for arbitrary subsets of the ordinary square grid, and is inspired by recent constructions for the Erd\\H{o}s unit distance"},"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":"2607.05374","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-06T17:52:52Z","cross_cats_sorted":["math.MG","math.NT"],"title_canon_sha256":"7bd12c7baa1eeb6cfe64ded0e7fe61fec02ca3fb1172e96a493c0380434ad1db","abstract_canon_sha256":"9b22ddf67d1d9ecc146797da11a8c1e3bf65cd7b8b1ad7b3b013292acfad01d8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-07T03:19:25.288653Z","signature_b64":"kX9eEfYpNeJM0pxt909lW2klIWL6p41TPk7nLef5IgR58vQJAQNFwd3ZU0QALnmyxAGZX6XU2n/V2xfyWaKDDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c50f35432fc99f529f7e0905e932bc69d27ff87cc25bd96dbd923a998d865527","last_reissued_at":"2026-07-07T03:19:25.288207Z","signature_status":"signed_v1","first_computed_at":"2026-07-07T03:19:25.288207Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Minkowski grid has robustly many repeated distances","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG","math.NT"],"primary_cat":"math.CO","authors_text":"Cosmin Pohoata, Daniel G. Zhu, Sungchul Lee","submitted_at":"2026-07-06T17:52:52Z","abstract_excerpt":"We show that there exists a constant $\\delta > 0$ such that for any positive integer $n$ there exists a set of $n$ points $P \\subset \\mathbb{R}^2$ with the following property: for every subset $A \\subseteq P$ of size $|A| \\geq 2$, \\[ \\max_{\\lambda>0}\n  \\#\\{(a,b)\\in A \\times A:\n  a\\ne b,\\ \\lvert a-b\\rvert=\\lambda\\} \\gtrsim \\frac{|A|^2}{n^{1-\\delta}}.\\] Our result is a vertical amplification of a robust Ramanujan estimate recently established by Croot-Mao-Pohoata-Sheffer-Yip for arbitrary subsets of the ordinary square grid, and is inspired by recent constructions for the Erd\\H{o}s unit distance"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.05374","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/2607.05374/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":"2607.05374","created_at":"2026-07-07T03:19:25.288269+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.05374v1","created_at":"2026-07-07T03:19:25.288269+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.05374","created_at":"2026-07-07T03:19:25.288269+00:00"},{"alias_kind":"pith_short_12","alias_value":"YUHTKQZPZGPV","created_at":"2026-07-07T03:19:25.288269+00:00"},{"alias_kind":"pith_short_16","alias_value":"YUHTKQZPZGPVFH36","created_at":"2026-07-07T03:19:25.288269+00:00"},{"alias_kind":"pith_short_8","alias_value":"YUHTKQZP","created_at":"2026-07-07T03:19:25.288269+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/YUHTKQZPZGPVFH36BEC6SMV4NH","json":"https://pith.science/pith/YUHTKQZPZGPVFH36BEC6SMV4NH.json","graph_json":"https://pith.science/api/pith-number/YUHTKQZPZGPVFH36BEC6SMV4NH/graph.json","events_json":"https://pith.science/api/pith-number/YUHTKQZPZGPVFH36BEC6SMV4NH/events.json","paper":"https://pith.science/paper/YUHTKQZP"},"agent_actions":{"view_html":"https://pith.science/pith/YUHTKQZPZGPVFH36BEC6SMV4NH","download_json":"https://pith.science/pith/YUHTKQZPZGPVFH36BEC6SMV4NH.json","view_paper":"https://pith.science/paper/YUHTKQZP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.05374&json=true","fetch_graph":"https://pith.science/api/pith-number/YUHTKQZPZGPVFH36BEC6SMV4NH/graph.json","fetch_events":"https://pith.science/api/pith-number/YUHTKQZPZGPVFH36BEC6SMV4NH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YUHTKQZPZGPVFH36BEC6SMV4NH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YUHTKQZPZGPVFH36BEC6SMV4NH/action/storage_attestation","attest_author":"https://pith.science/pith/YUHTKQZPZGPVFH36BEC6SMV4NH/action/author_attestation","sign_citation":"https://pith.science/pith/YUHTKQZPZGPVFH36BEC6SMV4NH/action/citation_signature","submit_replication":"https://pith.science/pith/YUHTKQZPZGPVFH36BEC6SMV4NH/action/replication_record"}},"created_at":"2026-07-07T03:19:25.288269+00:00","updated_at":"2026-07-07T03:19:25.288269+00:00"}