{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:7IBLZBI6MIHHEQYU44XBHVDSW4","short_pith_number":"pith:7IBLZBI6","canonical_record":{"source":{"id":"1008.3671","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-08-22T00:37:44Z","cross_cats_sorted":[],"title_canon_sha256":"6f5e9cef1da371f52231cd87870229542db676c6972e9c4e79ef63736412a5d4","abstract_canon_sha256":"eadd0da4aa8d12c757034ce364f39f26889c0ad63a35cbb6bc50b90a25566e47"},"schema_version":"1.0"},"canonical_sha256":"fa02bc851e620e724314e72e13d472b729cd3af2591bfb2bb71ba650a58f394e","source":{"kind":"arxiv","id":"1008.3671","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.3671","created_at":"2026-05-18T02:53:54Z"},{"alias_kind":"arxiv_version","alias_value":"1008.3671v2","created_at":"2026-05-18T02:53:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.3671","created_at":"2026-05-18T02:53:54Z"},{"alias_kind":"pith_short_12","alias_value":"7IBLZBI6MIHH","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"7IBLZBI6MIHHEQYU","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"7IBLZBI6","created_at":"2026-05-18T12:26:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:7IBLZBI6MIHHEQYU44XBHVDSW4","target":"record","payload":{"canonical_record":{"source":{"id":"1008.3671","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-08-22T00:37:44Z","cross_cats_sorted":[],"title_canon_sha256":"6f5e9cef1da371f52231cd87870229542db676c6972e9c4e79ef63736412a5d4","abstract_canon_sha256":"eadd0da4aa8d12c757034ce364f39f26889c0ad63a35cbb6bc50b90a25566e47"},"schema_version":"1.0"},"canonical_sha256":"fa02bc851e620e724314e72e13d472b729cd3af2591bfb2bb71ba650a58f394e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:53:54.498158Z","signature_b64":"L39QOqbKEzfy4ODLls3JFcjbo192ZX3lggFKxwZ73GIiqKq+3Gu9ts48ACvR9XZsrDWhrCcrwZOPVE7vHYo1Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fa02bc851e620e724314e72e13d472b729cd3af2591bfb2bb71ba650a58f394e","last_reissued_at":"2026-05-18T02:53:54.497302Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:53:54.497302Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1008.3671","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:53:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZmRBzIeHPEdPjzBLtNcYIvLWd6/FA1TrUntiCsZEbxDUuKjusXWVDBbKgULDwnsw/3/CBme5xZZICIGT2z13Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T02:56:26.807768Z"},"content_sha256":"91ebf4db2f33f5b64b009e13e32394b24156a29aa257597a8251d972186877ae","schema_version":"1.0","event_id":"sha256:91ebf4db2f33f5b64b009e13e32394b24156a29aa257597a8251d972186877ae"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:7IBLZBI6MIHHEQYU44XBHVDSW4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Rational Distances with Rational Angles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Frank de Zeeuw, J\\'ozsef Solymosi, Ryan Schwartz","submitted_at":"2010-08-22T00:37:44Z","abstract_excerpt":"In 1946 Erd\\H os asked for the maximum number of unit distances, $u(n)$, among $n$ points in the plane. He showed that $u(n)> n^{1+c/\\log\\log n}$ and conjectured that this was the true magnitude. The best known upper bound is $u(n)<cn^{4/3}$, due to Spencer, Szemer\\'edi and Trotter. We show that the upper bound $n^{1+6/\\sqrt{\\log n}}$ holds if we only consider unit distances with rational angle, by which we mean that the line through the pair of points makes a rational angle in degrees with the x-axis. Using an algebraic theorem of Mann we get a uniform bound on the number of paths between two"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.3671","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:53:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zcP2CwZCY+aqCldXf0Y68WABP5/RR53QCvDHRwb69U8c8FPXNAezI/jvl5Y6cDWnbHLiDRq6V/3HdYlrG6LvBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T02:56:26.808466Z"},"content_sha256":"44d1e4f414105555b52bdc6aa60cd7a26e31bf3c5d5ae128a70f5e44d4b6376f","schema_version":"1.0","event_id":"sha256:44d1e4f414105555b52bdc6aa60cd7a26e31bf3c5d5ae128a70f5e44d4b6376f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7IBLZBI6MIHHEQYU44XBHVDSW4/bundle.json","state_url":"https://pith.science/pith/7IBLZBI6MIHHEQYU44XBHVDSW4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7IBLZBI6MIHHEQYU44XBHVDSW4/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-08T02:56:26Z","links":{"resolver":"https://pith.science/pith/7IBLZBI6MIHHEQYU44XBHVDSW4","bundle":"https://pith.science/pith/7IBLZBI6MIHHEQYU44XBHVDSW4/bundle.json","state":"https://pith.science/pith/7IBLZBI6MIHHEQYU44XBHVDSW4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7IBLZBI6MIHHEQYU44XBHVDSW4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:7IBLZBI6MIHHEQYU44XBHVDSW4","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":"eadd0da4aa8d12c757034ce364f39f26889c0ad63a35cbb6bc50b90a25566e47","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-08-22T00:37:44Z","title_canon_sha256":"6f5e9cef1da371f52231cd87870229542db676c6972e9c4e79ef63736412a5d4"},"schema_version":"1.0","source":{"id":"1008.3671","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.3671","created_at":"2026-05-18T02:53:54Z"},{"alias_kind":"arxiv_version","alias_value":"1008.3671v2","created_at":"2026-05-18T02:53:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.3671","created_at":"2026-05-18T02:53:54Z"},{"alias_kind":"pith_short_12","alias_value":"7IBLZBI6MIHH","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"7IBLZBI6MIHHEQYU","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"7IBLZBI6","created_at":"2026-05-18T12:26:04Z"}],"graph_snapshots":[{"event_id":"sha256:44d1e4f414105555b52bdc6aa60cd7a26e31bf3c5d5ae128a70f5e44d4b6376f","target":"graph","created_at":"2026-05-18T02:53:54Z","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"},"paper":{"abstract_excerpt":"In 1946 Erd\\H os asked for the maximum number of unit distances, $u(n)$, among $n$ points in the plane. He showed that $u(n)> n^{1+c/\\log\\log n}$ and conjectured that this was the true magnitude. The best known upper bound is $u(n)<cn^{4/3}$, due to Spencer, Szemer\\'edi and Trotter. We show that the upper bound $n^{1+6/\\sqrt{\\log n}}$ holds if we only consider unit distances with rational angle, by which we mean that the line through the pair of points makes a rational angle in degrees with the x-axis. Using an algebraic theorem of Mann we get a uniform bound on the number of paths between two","authors_text":"Frank de Zeeuw, J\\'ozsef Solymosi, Ryan Schwartz","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-08-22T00:37:44Z","title":"Rational Distances with Rational Angles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.3671","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":"sha256:91ebf4db2f33f5b64b009e13e32394b24156a29aa257597a8251d972186877ae","target":"record","created_at":"2026-05-18T02:53:54Z","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":"eadd0da4aa8d12c757034ce364f39f26889c0ad63a35cbb6bc50b90a25566e47","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-08-22T00:37:44Z","title_canon_sha256":"6f5e9cef1da371f52231cd87870229542db676c6972e9c4e79ef63736412a5d4"},"schema_version":"1.0","source":{"id":"1008.3671","kind":"arxiv","version":2}},"canonical_sha256":"fa02bc851e620e724314e72e13d472b729cd3af2591bfb2bb71ba650a58f394e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fa02bc851e620e724314e72e13d472b729cd3af2591bfb2bb71ba650a58f394e","first_computed_at":"2026-05-18T02:53:54.497302Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:53:54.497302Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"L39QOqbKEzfy4ODLls3JFcjbo192ZX3lggFKxwZ73GIiqKq+3Gu9ts48ACvR9XZsrDWhrCcrwZOPVE7vHYo1Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:53:54.498158Z","signed_message":"canonical_sha256_bytes"},"source_id":"1008.3671","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:91ebf4db2f33f5b64b009e13e32394b24156a29aa257597a8251d972186877ae","sha256:44d1e4f414105555b52bdc6aa60cd7a26e31bf3c5d5ae128a70f5e44d4b6376f"],"state_sha256":"acfd6aaf3078deca598ddcbdedbc0c7b17f41b9b3762203b56508b2693df88f6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uON5h7b7uNZc4FgdusOHHx48x/sr4nzLRs1bIB6y9R41Xp6ynN9joBQuVZWBLDYh3zgHZ3wmn8vXlbPE/E8fAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T02:56:26.812764Z","bundle_sha256":"b244698902d7010322b176e3f3e16d8891973241675e0cd6754f04ffbedee23e"}}