{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:IKMTDNKCM5C4PFVH3BFXDLYYJK","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":"3f71adb459971c53c522305f2d11637c64dd4c23c173e6fde9f39cea646370ed","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-12-12T18:57:24Z","title_canon_sha256":"cbf29f7130f06d58ebca37ec653c21158acda71a691fa0dbacd04450e3c5d7b5"},"schema_version":"1.0","source":{"id":"1212.2930","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.2930","created_at":"2026-05-18T03:38:34Z"},{"alias_kind":"arxiv_version","alias_value":"1212.2930v1","created_at":"2026-05-18T03:38:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.2930","created_at":"2026-05-18T03:38:34Z"},{"alias_kind":"pith_short_12","alias_value":"IKMTDNKCM5C4","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"IKMTDNKCM5C4PFVH","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"IKMTDNKC","created_at":"2026-05-18T12:27:09Z"}],"graph_snapshots":[{"event_id":"sha256:9a0c18e3fcbd0260850aed15203c0d3c3a3f1831dc05c6607b7ed918aae51054","target":"graph","created_at":"2026-05-18T03:38:34Z","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":"Many problems in additive number theory, such as Fermat's last theorem and the twin prime conjecture, can be understood by examining sums or differences of a set with itself. A finite set $A \\subset \\mathbb{Z}$ is considered sum-dominant if $|A+A|>|A-A|$. If we consider all subsets of ${0, 1, ..., n-1}$, as $n\\to\\infty$ it is natural to expect that almost all subsets should be difference-dominant, as addition is commutative but subtraction is not; however, Martin and O'Bryant in 2007 proved that a positive percentage are sum-dominant as $n\\to\\infty$.\n  This motivates the study of \"coordinate s","authors_text":"Amanda Bower, Ron Evans, Steven J. Miller, Victor Luo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-12-12T18:57:24Z","title":"Coordinate sum and difference sets of $d$-dimensional modular hyperbolas"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.2930","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:d0a0e624f7a15198994dafc2f9aec16ea5b7ccf698f8c080d3875ef0726d6a69","target":"record","created_at":"2026-05-18T03:38:34Z","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":"3f71adb459971c53c522305f2d11637c64dd4c23c173e6fde9f39cea646370ed","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-12-12T18:57:24Z","title_canon_sha256":"cbf29f7130f06d58ebca37ec653c21158acda71a691fa0dbacd04450e3c5d7b5"},"schema_version":"1.0","source":{"id":"1212.2930","kind":"arxiv","version":1}},"canonical_sha256":"429931b5426745c796a7d84b71af184a89d46ed11252c9b764236a067e7a8b7c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"429931b5426745c796a7d84b71af184a89d46ed11252c9b764236a067e7a8b7c","first_computed_at":"2026-05-18T03:38:34.905707Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:38:34.905707Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"FSCFpHpzpWuZgfj0dTO7VidWo2YjMpDsFAvJl/y6dXMjAmOjbBwER8nD5XlSzJvIwbwKCxYqUGHKHKVR82bjDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:38:34.906294Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.2930","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d0a0e624f7a15198994dafc2f9aec16ea5b7ccf698f8c080d3875ef0726d6a69","sha256:9a0c18e3fcbd0260850aed15203c0d3c3a3f1831dc05c6607b7ed918aae51054"],"state_sha256":"56dcc02669d67e6dda8dadb912297186e1f5004e3a809f152ecacdc9e20c7665"}