{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:Q3DEY5O4FDHXUI5QSQXYLLOE2G","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":"4cf83bfec407a175d57d4b24a4b84b0a512d0c5963a517d6d744ebf3da6376a0","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-09-30T13:16:44Z","title_canon_sha256":"4a0d55a2eafad6df4d085358c10f4b1946721f17213ee33fa958a028b9298206"},"schema_version":"1.0","source":{"id":"1409.8535","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.8535","created_at":"2026-05-18T00:56:46Z"},{"alias_kind":"arxiv_version","alias_value":"1409.8535v2","created_at":"2026-05-18T00:56:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.8535","created_at":"2026-05-18T00:56:46Z"},{"alias_kind":"pith_short_12","alias_value":"Q3DEY5O4FDHX","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_16","alias_value":"Q3DEY5O4FDHXUI5Q","created_at":"2026-05-18T12:28:43Z"},{"alias_kind":"pith_short_8","alias_value":"Q3DEY5O4","created_at":"2026-05-18T12:28:43Z"}],"graph_snapshots":[{"event_id":"sha256:7ae86ad2f2715146d7e86eac78006641222e0000ed4f81d5deea20943aa866d2","target":"graph","created_at":"2026-05-18T00:56:46Z","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":"An order-preserving Freiman 2-isomorphism is a map $\\phi:X \\rightarrow \\mathbb{R}$ such that $\\phi(a) < \\phi(b)$ if and only if $a < b$ and $\\phi(a)+\\phi(b) = \\phi(c)+\\phi(d)$ if and only if $a+b=c+d$ for any $a,b,c,d \\in X$. We show that for any $A \\subseteq \\mathbb{Z}$, if $|A+A| \\le K|A|$, then there exists a subset $A' \\subseteq A$ such that the following holds: $|A'| \\gg_K |A|$ and there exists an order-preserving Freiman 2-isomorphism $\\phi: A' \\rightarrow [-c|A|,c|A|] \\cap \\mathbb{Z}$ where $c$ depends only on $K$. Several applications are also presented.","authors_text":"Albert Bush, Ernie Croot, Gagik Amirkhanyan","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-09-30T13:16:44Z","title":"Order-preserving Freiman isomorphisms"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.8535","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:a563d34397030a3a1f0feb71b593374532f2e3ad66d4730b44760f87935d15fe","target":"record","created_at":"2026-05-18T00:56:46Z","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":"4cf83bfec407a175d57d4b24a4b84b0a512d0c5963a517d6d744ebf3da6376a0","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-09-30T13:16:44Z","title_canon_sha256":"4a0d55a2eafad6df4d085358c10f4b1946721f17213ee33fa958a028b9298206"},"schema_version":"1.0","source":{"id":"1409.8535","kind":"arxiv","version":2}},"canonical_sha256":"86c64c75dc28cf7a23b0942f85adc4d1a58ad67505ddb43b76351118f1ded135","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"86c64c75dc28cf7a23b0942f85adc4d1a58ad67505ddb43b76351118f1ded135","first_computed_at":"2026-05-18T00:56:46.927791Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:56:46.927791Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"R5WO9MaK5k/lqciifblKVC3RH/C3X+k344NkMx64+j+B/GzmCLOefsC35yVeU3Un+xDg+28G90OM5HIEIx/aAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:56:46.928449Z","signed_message":"canonical_sha256_bytes"},"source_id":"1409.8535","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a563d34397030a3a1f0feb71b593374532f2e3ad66d4730b44760f87935d15fe","sha256:7ae86ad2f2715146d7e86eac78006641222e0000ed4f81d5deea20943aa866d2"],"state_sha256":"0cbc72554b8146a5289f78557854154367597861cc88076b78330d60f6ceed9a"}