{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:FFCBZX6MRVKTSVRH7MZFZXRGV7","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":"8a1ffc8bf513074848ec2ffe22f0ce04c4f0e0815ebc03c6b843bbd33b4ead0f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-08-26T21:00:52Z","title_canon_sha256":"a3ef2e08c0a3610a51df96aac8bee5456a2a28725f4ca2f8d9c082b10020f0d2"},"schema_version":"1.0","source":{"id":"2108.12034","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2108.12034","created_at":"2026-07-05T05:06:52Z"},{"alias_kind":"arxiv_version","alias_value":"2108.12034v2","created_at":"2026-07-05T05:06:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2108.12034","created_at":"2026-07-05T05:06:52Z"},{"alias_kind":"pith_short_12","alias_value":"FFCBZX6MRVKT","created_at":"2026-07-05T05:06:52Z"},{"alias_kind":"pith_short_16","alias_value":"FFCBZX6MRVKTSVRH","created_at":"2026-07-05T05:06:52Z"},{"alias_kind":"pith_short_8","alias_value":"FFCBZX6M","created_at":"2026-07-05T05:06:52Z"}],"graph_snapshots":[{"event_id":"sha256:e68cc731b074ffc6ff2e715d37334b6fb758f0c726bbb6826111d5604cc0821d","target":"graph","created_at":"2026-07-05T05:06:52Z","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/2108.12034/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We characterize the largest point sets in the plane which define at most 1, 2, and 3 angles. For $P(k)$ the largest size of a point set admitting at most $k$ angles, we prove $P(2)=5$ and $P(3)=5$. We also provide the general bounds of $k+2 \\leq P(k) \\leq 6k$, although the upper bound may be improved pending progress toward the Weak Dirac Conjecture. Notably, it is surprising that $P(k)=\\Theta(k)$ since, in the distance setting, the best known upper bound on the analogous quantity is quadratic and no lower bound is well-understood.","authors_text":"Charles Wolf, Ethan Pesikoff, Eyvindur A. Palsson, Henry L. Fleischmann, Steven J. Miller","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-08-26T21:00:52Z","title":"Optimal Point Sets Determining Few Distinct Angles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.12034","kind":"arxiv","version":2},"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":"sha256:6ff31ba23694fafafac5229fc8ff573086c3c1763fca55f38d2249b16c13f55f","target":"record","created_at":"2026-07-05T05:06:52Z","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":"8a1ffc8bf513074848ec2ffe22f0ce04c4f0e0815ebc03c6b843bbd33b4ead0f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2021-08-26T21:00:52Z","title_canon_sha256":"a3ef2e08c0a3610a51df96aac8bee5456a2a28725f4ca2f8d9c082b10020f0d2"},"schema_version":"1.0","source":{"id":"2108.12034","kind":"arxiv","version":2}},"canonical_sha256":"29441cdfcc8d55395627fb325cde26afd755ee2b6fa7c89ee45e88c670ffa504","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"29441cdfcc8d55395627fb325cde26afd755ee2b6fa7c89ee45e88c670ffa504","first_computed_at":"2026-07-05T05:06:52.046915Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:06:52.046915Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"j2Fqu7amLIXA0nnmsfDRWiTKr2RdrfcD3ZxsqsltcqyfWI4fIssHVaplqUmdwIwrxvg+JuPtJYPVA1Yq9UOWBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T05:06:52.047274Z","signed_message":"canonical_sha256_bytes"},"source_id":"2108.12034","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6ff31ba23694fafafac5229fc8ff573086c3c1763fca55f38d2249b16c13f55f","sha256:e68cc731b074ffc6ff2e715d37334b6fb758f0c726bbb6826111d5604cc0821d"],"state_sha256":"f5982b752ef8e305dc17b0903b5547aebdf520f0b902f578099f1f974af98f1d"}