{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:UJZOPEI566BNKFYE4LLIAFM5E5","short_pith_number":"pith:UJZOPEI5","canonical_record":{"source":{"id":"1503.09071","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-03-26T20:33:35Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"fd2717339a85410271c55a90e10432bb12c1fc9f6d6600848bd13c3c4dc8e087","abstract_canon_sha256":"c47779453f7e29cbcda94c602f07e885909139e5e6f6f7f310b5768e45c60304"},"schema_version":"1.0"},"canonical_sha256":"a272e7911df782d51704e2d680159d27619de6e746ab467658f324168947222e","source":{"kind":"arxiv","id":"1503.09071","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.09071","created_at":"2026-05-18T01:36:47Z"},{"alias_kind":"arxiv_version","alias_value":"1503.09071v1","created_at":"2026-05-18T01:36:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.09071","created_at":"2026-05-18T01:36:47Z"},{"alias_kind":"pith_short_12","alias_value":"UJZOPEI566BN","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"UJZOPEI566BNKFYE","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"UJZOPEI5","created_at":"2026-05-18T12:29:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:UJZOPEI566BNKFYE4LLIAFM5E5","target":"record","payload":{"canonical_record":{"source":{"id":"1503.09071","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-03-26T20:33:35Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"fd2717339a85410271c55a90e10432bb12c1fc9f6d6600848bd13c3c4dc8e087","abstract_canon_sha256":"c47779453f7e29cbcda94c602f07e885909139e5e6f6f7f310b5768e45c60304"},"schema_version":"1.0"},"canonical_sha256":"a272e7911df782d51704e2d680159d27619de6e746ab467658f324168947222e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:47.908424Z","signature_b64":"3wWpKW97tcOVY5GdoZGpvh6egsFY+2NOKnNYLWiHsbDS53JZqePTUfhpGTVDTyPQzZl7qlG3bYAoP9rSecsRAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a272e7911df782d51704e2d680159d27619de6e746ab467658f324168947222e","last_reissued_at":"2026-05-18T01:36:47.907862Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:47.907862Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1503.09071","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:36:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fudUrfyw1yr/wHHC/xTcOHUOZ+vC1FKj0+u+ENpOSCR1UXlOIps+a7NTcnlMJNmab4B9ac7MSLNH0GrcJ4AiBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T02:22:27.541201Z"},"content_sha256":"abf7580b9ee737b337c38294c43dbcfdf3bfa5f9918ac1d2cd3e748e2fb7e72a","schema_version":"1.0","event_id":"sha256:abf7580b9ee737b337c38294c43dbcfdf3bfa5f9918ac1d2cd3e748e2fb7e72a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:UJZOPEI566BNKFYE4LLIAFM5E5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Exact Kronecker Constants of Three Element Sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CA","authors_text":"Kathryn E. Hare, L. Thomas Ramsey","submitted_at":"2015-03-26T20:33:35Z","abstract_excerpt":"For any three element set of positive integers, $\\{a,b,n\\}$, with $a<b<n$, $n$ sufficiently large and $\\gcd(a,b)=1$, we find the least $\\alpha$ such that given any real numbers $t_1$, $t_2$, $t_3$, there is a real number $x$ such that \\begin{equation*} \\max \\{\\left\\langle ax-t_{1}\\right\\rangle ,\\left\\langle bx-t_{2}\\right\\rangle ,\\left\\langle nx-t_{3}\\right\\rangle \\}\\leq \\alpha , \\end{equation*} where $\\left\\langle \\cdot \\right\\rangle $ denotes the distance to the nearest integer. The number $\\alpha $ is known as the angular Kronecker constant of $\\{a,b,n\\}$. We also find the least $\\beta $ su"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.09071","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":""},"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":"2026-05-18T01:36:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v6O78nIXzJJlPz8yKmNSypIhvc1C4UwfOWT/cVtR5a2p3DqIvAIa6b0a31pSnBUVFFkn51/cWFeJyGiLlBpLBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-12T02:22:27.541965Z"},"content_sha256":"79b5d409b91fe1167c9b32970e7a7ae30431360bc6b5fa4355fdd766f32bb994","schema_version":"1.0","event_id":"sha256:79b5d409b91fe1167c9b32970e7a7ae30431360bc6b5fa4355fdd766f32bb994"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UJZOPEI566BNKFYE4LLIAFM5E5/bundle.json","state_url":"https://pith.science/pith/UJZOPEI566BNKFYE4LLIAFM5E5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UJZOPEI566BNKFYE4LLIAFM5E5/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":"2026-06-12T02:22:27Z","links":{"resolver":"https://pith.science/pith/UJZOPEI566BNKFYE4LLIAFM5E5","bundle":"https://pith.science/pith/UJZOPEI566BNKFYE4LLIAFM5E5/bundle.json","state":"https://pith.science/pith/UJZOPEI566BNKFYE4LLIAFM5E5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UJZOPEI566BNKFYE4LLIAFM5E5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:UJZOPEI566BNKFYE4LLIAFM5E5","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":"c47779453f7e29cbcda94c602f07e885909139e5e6f6f7f310b5768e45c60304","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-03-26T20:33:35Z","title_canon_sha256":"fd2717339a85410271c55a90e10432bb12c1fc9f6d6600848bd13c3c4dc8e087"},"schema_version":"1.0","source":{"id":"1503.09071","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.09071","created_at":"2026-05-18T01:36:47Z"},{"alias_kind":"arxiv_version","alias_value":"1503.09071v1","created_at":"2026-05-18T01:36:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.09071","created_at":"2026-05-18T01:36:47Z"},{"alias_kind":"pith_short_12","alias_value":"UJZOPEI566BN","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_16","alias_value":"UJZOPEI566BNKFYE","created_at":"2026-05-18T12:29:44Z"},{"alias_kind":"pith_short_8","alias_value":"UJZOPEI5","created_at":"2026-05-18T12:29:44Z"}],"graph_snapshots":[{"event_id":"sha256:79b5d409b91fe1167c9b32970e7a7ae30431360bc6b5fa4355fdd766f32bb994","target":"graph","created_at":"2026-05-18T01:36:47Z","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":"For any three element set of positive integers, $\\{a,b,n\\}$, with $a<b<n$, $n$ sufficiently large and $\\gcd(a,b)=1$, we find the least $\\alpha$ such that given any real numbers $t_1$, $t_2$, $t_3$, there is a real number $x$ such that \\begin{equation*} \\max \\{\\left\\langle ax-t_{1}\\right\\rangle ,\\left\\langle bx-t_{2}\\right\\rangle ,\\left\\langle nx-t_{3}\\right\\rangle \\}\\leq \\alpha , \\end{equation*} where $\\left\\langle \\cdot \\right\\rangle $ denotes the distance to the nearest integer. The number $\\alpha $ is known as the angular Kronecker constant of $\\{a,b,n\\}$. We also find the least $\\beta $ su","authors_text":"Kathryn E. Hare, L. Thomas Ramsey","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-03-26T20:33:35Z","title":"Exact Kronecker Constants of Three Element Sets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.09071","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:abf7580b9ee737b337c38294c43dbcfdf3bfa5f9918ac1d2cd3e748e2fb7e72a","target":"record","created_at":"2026-05-18T01:36:47Z","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":"c47779453f7e29cbcda94c602f07e885909139e5e6f6f7f310b5768e45c60304","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-03-26T20:33:35Z","title_canon_sha256":"fd2717339a85410271c55a90e10432bb12c1fc9f6d6600848bd13c3c4dc8e087"},"schema_version":"1.0","source":{"id":"1503.09071","kind":"arxiv","version":1}},"canonical_sha256":"a272e7911df782d51704e2d680159d27619de6e746ab467658f324168947222e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a272e7911df782d51704e2d680159d27619de6e746ab467658f324168947222e","first_computed_at":"2026-05-18T01:36:47.907862Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:47.907862Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"3wWpKW97tcOVY5GdoZGpvh6egsFY+2NOKnNYLWiHsbDS53JZqePTUfhpGTVDTyPQzZl7qlG3bYAoP9rSecsRAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:47.908424Z","signed_message":"canonical_sha256_bytes"},"source_id":"1503.09071","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:abf7580b9ee737b337c38294c43dbcfdf3bfa5f9918ac1d2cd3e748e2fb7e72a","sha256:79b5d409b91fe1167c9b32970e7a7ae30431360bc6b5fa4355fdd766f32bb994"],"state_sha256":"16660d2c0d16ab5f5eb1ecd17410bdcb29c2ba5d9554dee2d058d1cbad4634db"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZMeWcRIT6tnZo0/UGTJJui75c1StkS/tAeef9EDeWgWfRQ1QLgA6XhKVOu64FkAyTvEtk7vtGFaI/pb4FEj8Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-12T02:22:27.546186Z","bundle_sha256":"ae645f359204eebc846a2733008c1730e120fcb5a3aa15a849062ad6ddf91a25"}}