{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:YUHTKQZPZGPVFH36BEC6SMV4NH","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"9b22ddf67d1d9ecc146797da11a8c1e3bf65cd7b8b1ad7b3b013292acfad01d8","cross_cats_sorted":["math.MG","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-06T17:52:52Z","title_canon_sha256":"7bd12c7baa1eeb6cfe64ded0e7fe61fec02ca3fb1172e96a493c0380434ad1db"},"schema_version":"1.0","source":{"id":"2607.05374","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.05374","created_at":"2026-07-07T03:19:25Z"},{"alias_kind":"arxiv_version","alias_value":"2607.05374v1","created_at":"2026-07-07T03:19:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.05374","created_at":"2026-07-07T03:19:25Z"},{"alias_kind":"pith_short_12","alias_value":"YUHTKQZPZGPV","created_at":"2026-07-07T03:19:25Z"},{"alias_kind":"pith_short_16","alias_value":"YUHTKQZPZGPVFH36","created_at":"2026-07-07T03:19:25Z"},{"alias_kind":"pith_short_8","alias_value":"YUHTKQZP","created_at":"2026-07-07T03:19:25Z"}],"graph_snapshots":[{"event_id":"sha256:5b1d028c48210e5d00e08853ed615f4e81d6b9201ae98daf3a95393662e57fa6","target":"graph","created_at":"2026-07-07T03:19:25Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2607.05374/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Cosmin Pohoata, Daniel G. Zhu, Sungchul Lee","cross_cats":["math.MG","math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-06T17:52:52Z","title":"The Minkowski grid has robustly many repeated distances"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.05374","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:6e3dd5313007a4a43ad60e8a1ab008ad64ed97da8bd204c781a8687acba57278","target":"record","created_at":"2026-07-07T03:19:25Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"9b22ddf67d1d9ecc146797da11a8c1e3bf65cd7b8b1ad7b3b013292acfad01d8","cross_cats_sorted":["math.MG","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-06T17:52:52Z","title_canon_sha256":"7bd12c7baa1eeb6cfe64ded0e7fe61fec02ca3fb1172e96a493c0380434ad1db"},"schema_version":"1.0","source":{"id":"2607.05374","kind":"arxiv","version":1}},"canonical_sha256":"c50f35432fc99f529f7e0905e932bc69d27ff87cc25bd96dbd923a998d865527","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c50f35432fc99f529f7e0905e932bc69d27ff87cc25bd96dbd923a998d865527","first_computed_at":"2026-07-07T03:19:25.288207Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-07T03:19:25.288207Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kX9eEfYpNeJM0pxt909lW2klIWL6p41TPk7nLef5IgR58vQJAQNFwd3ZU0QALnmyxAGZX6XU2n/V2xfyWaKDDw==","signature_status":"signed_v1","signed_at":"2026-07-07T03:19:25.288653Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.05374","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6e3dd5313007a4a43ad60e8a1ab008ad64ed97da8bd204c781a8687acba57278","sha256:5b1d028c48210e5d00e08853ed615f4e81d6b9201ae98daf3a95393662e57fa6"],"state_sha256":"65e82aa176b83b5ecd152215928755e1ef9d06a66f6f6a5ae69a6342040233ea"}